UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


*t«t-*r 


' 


A 


NOTES 


DESIGN  OF  MACHINE  ELEMENTS 


FOR  USE  IN  CONNECTION  WITH  UNWIN'S 
MACHINE  DESIGN,  PART  I. 


BY  JOHN  H.  BARR, 

Professor  of  Machine  Design,  Sibley  College, 
Cornell  University. 


ITHACA,    NEW   YORK, 
1901. 


ANDRUS  &  CHURCH,  PRINTERS. 


TJ 


PREFACE. 


These  notes  were  prepared  to  accompany  Professor  W.  C. 
Uuwin's  Elements  of  Machine  Design,  Part  I.  Taken  in  con- 
nection with  this  text-book,  they  form  an  outline  of  the  course  in 
the  Design  of  Machine  Elements  as  given  to  the  Junior  class  of 
Sibley  College,  Cornell  University. 

The  arrangement  of  the  topics  indicates  the  order  in  which  the 
subjects  are  discussed,  so  that  these  notes  serve  as  a  syllabus  of 
the  lectures  as  well  as  a  commentary  on  the  text-book.  In  order 
that  this  double  function  may  be  fulfilled,  numerous  headings  of 
articles  are  inserted  accompanied  simply  by  references  to  Pro- 
fessor Unwin's  book.  When  it  has  seemed  desirable  to  supple- 
ment or  qualify  the  statements  of  the  text-book,  comments  follow 
the  appropriate  references.  The  treatment  of  certain  topics  is 
quite  independent  of  the  text-book,  and  references  to  the  authori- 
ties used  are  generally  given  in  connection  with  the  discussion  of 
such  subjects. 

A  short  list  of  Reference  Books  is  added  to  suggest  the  sources 
of  fuller  information   and  data.     These  books  are  arranged    in 
K  classes,  as  an  indication  of  their  general  scope,  but  the3'  overlap 
\J    to  a  considerable  degree. 

L          The  preparation  of  these  Notes  has  extended  over  a  period  of 
^    two  or  three  years,  advanced  sheets  having  been  printed  and  dis- 
tributed to  the  classes  from  time  to  time.      The  conditions  under 
which  they  have  been  issued  has  necessarily  resulted  in  errors 
and  imperfections,  and  many  of  these  are  apparent. 

I  desire  to  acknowledge  my  great  obligation  to  the  numerous 
writers  and  investigators  consulted.  I  am  especially  indebted  to 
Professor  Dexter  S.  Kimball  and  Mr.  William  N.  Barnard*  for 
their  helpful  criticism  and  careful  reading  of  the  manuscript  and 
proof. 

JOHN  H.  BARR. 
Ithaca,  New  York, 
March,  1901. 


-209403 


REFERENCE    BOOKS. 

Materials  of  Engineering. 

Materials  of  Construction R.  H.  Thurston 

Materials  of  Construction J.  B.  Johnson 

Mechanics  of  Engineering. 

Mechanics  of  Engineering I.  P.  Church 

Mechanics  of  Materials  . Mansfield  Merriman 

Applied  Mechanics J.  H.  Cotterill 

Mechanics  of  Machinery A.  B.  W.  Kennedy 

Machinery  and  Millwork W.  J.  M.  Rankine 

General  Design. 

The  Constructor F.   Reuleaux 

Machine  Design A.  W.  Smith 

Machine  Design J.  F.  Klein 

Machine  Design F.  R.  Jones 

Special  Subjects. 

Friction  and  Lost  Work R.  H.  Thurston 

Machinery  of  Transmission J.  Weisbach 

Gearing Brown  &  Sharp  Mfg.  Co.  (Beale) 

Kinematics,  or  Mechanical  Movements C.  W.  MacCord 

Teeth  of  Gears .'_.  ...George  B.  Grant 

Rope  Driving J.  J.  Flather 

Practice. 

Mechanical  Engineer's  Pocket-Book Wm.   Kent 

Mechanical  Engineer's  Pocket-Book D.  R.  Low 

Manual  of  Machine  Construction John  Richards 

Transactions  of  American  Society  of  Mechanical  Engineers. 
Transactions  of  American  Society  of  Civil  Engineers. 
Transactions  of  American  Institute  of  Electrical  Engineers. 
Transactions  of  American  Institute  of  Mining  Engineers. 
Transactions  of  Institution  of  Civil  Engineers  (Great  Britain). 
Transactions  of  Institution  of  Mechanical  Engineers  (Great  Britain). 
Engineering  Periodicals. 
Trade  Publications. 


I. 

STRAINING  ACTIONS  IN  MACHINES. 


1.  Forces    acting  on    Machine    Members. — [Unwin,  §  16, 
page  22  ] 

To  the  forces  specified  by  Unwin,  may  be  added  :  (7)  Magnetic 
attraction,  as  exerted  between  members  of  electrical  machines. 

2.  Nature    'of    Straining   Actions. — The    character    of    the 
straining   action  and  of  the  stress  which    results   from   a  given 
load  depends  upon  the  direction  and  point  of  application  of  the 
load  force,  (or  forces),  and  upon  the  form,  the  position,  and  the 
arrangement  of  the   supports,    of  the   member.     A   given    load 
may  produce  tension,  compression,  shearing,  flexure,  or  torsion  ; 
or  a  combination  of  these.     Of  course  tension  and  compression 
cannot  both  exist  at  the  same  time  between  any  pair  of  mole- 
cules, or  particles.      Flexure  is  a  combination  of  tensile  and  com- 
pressive  stresses  between  different  sets  of  molecules  ;  or,  as  it  is 
often  expressed,  in  different  fibres,  of  the  same  body.     Torsion  is 
a   special    form    of    shearing    stress.     Owing    to    the    frequent 
occurrence  of  flexure  and  torsion  it  is  convenient  to  treat  these  as 
elementary  forms  of  stress. 

The  stresses  due  to  tension,  compression  and  flexure  are  essen- 
tially molecular  actions  normal  to  the  planes  separating  adjacent 
sets  of  interacting  molecules  :  that  is,  the  stresses  increase  or  de- 
crease the  distances  between  these  molecules  along  lines  connect- 
ing them. 

The  primary  straining  effect  in  shearing  and  torsional  actions 
is  displacement  of  adjacent  molecules,  between  which  the  stress 
acts,  tangentially  to  the  planes  separating  such  molecules.  In 
uniform  shear,  the  interacting  molecules  move,  relatively,  with  a 
rectilinear  translation.  In  torsional.  action,  the  adjacent  mole- 
cules, each  side  of  the  plane  of  stress,  have  a  relative  rotation 
about  an  axis. 

3.  Ultimate  or  Breaking  Strength. — [Unwin,  §   17,  pages 
23-24  ;  also  Table  I,  pages  40-41.]     See,  also,  the  table  given  on 


i-~  <j  uo 

rO         CS 


"5 


!£Ii 


Q  CO   rO 

vfi  VO  CO 


Stress  at 
Elasticit 
Limit. 
Tension. 


O> 

VO 


0  0  0  0  0  0 


111 


-3  — 

page  2  ;  taken  from  Professor  A.  W.  Smith's  Constructive  Mate- 
rials of  Engineering. 

4.  General  Idea  of  the  Factor  of  Safety.— [Unwin,  §  17, 
pages  23-24  ] 

The  working  stress  in  a  member  must  be  less  than  the  ultimate 
strength  of  the  material,  because  : 

(a)  Members  of  structures  and  machines  are    not   made  to  be 
broken  in  ordinary  service. 

(b)  Materials  employed   in    engineering   usually    take    a   per- 
manent deformation,  or  set,  before  rupture  occurs. 

(c)  There  is  always  liability  of  defects  in  the   material  and  im- 
perfections in  workmanship. 

(d)  In   many  cases  there  is  danger  of  stress  greater  than  the 
normal  working  stress  from  an  occasional  excess  of  load,  or  from 
accidents  which  are  not  foreseen  or  computed  in  advance  of  their 
occurrence 

It  is  generally  essential  that  a  part  be  not  only  strong  enough 
to  avoid  breaking  under  the  regular  maximum  working  load,  but 
also  that  it  shall  not  receive  a  permanent  set  ;  for  a  machine 
member  ordinarily  becomes  useless  if  it  takes  such  set  after 
having  been  given  the  required  form.  In  many  cases  a  temporary 
strain,  even  considerably  below  the  elastic  limit,  would  seriously 
impair  the  accuracy  of  operation,  and  in  such  cases  the  members 
often  require  great  excess  of  strength  to  secure  sufficient  rigidity. 
It  follows  from  these  considerations  that  the  working  stress  should 
always  be  below  the  elastic  limit  and  it  must  often  be  much  lower 
than  the  elastic  strength 

The  elastic  strength  of  many  of  the  common  materials  of  con- 
struction is  not  much  above  one- half  the  ultimate  strength,  and 
the  proper  allowance  for  defects,  overloading  and  other  contingen- 
cies depends  upon  the  conditions  of  the  particular  case.  It  thus 
appears  that  the  working  stress  should  never  be  as  great  as  one- 
half,  and  it  should  seldom  exceed  one-third,  of  the  working 
strength  of  the  material  In  structures  liable  to  little  variation 
of  load  and  to  no  shock,  the  working  stress  may  be  from  one-third 
to  one  fourth  the  ultimate  strength,  with  such  comparatively 
homogeneous  and  ductile  materials  as  wrought  iron,  mild  steel, 


etc.  With  brittle  materials,  as  cast  iron,  hard  steel,  etc.  (which 
are  more  subject  to  hidden  defects  and  are  less  reliable  generally), 
a  greater  margin  is  required  for  safety.  If  the  conditions  are 
such  that  the  material  is  apt  to  deteriorate  seriously,  a  suita- 
ble decrease  of  computed  working  stress  should  be  made. 

The  effect  of  a  suddenly  applied  load  (shock  or  impact)  is  to 
produce  a  stress  in  excess  of  that  due  to  the  same  load  applied 
gradually,  and  where  such  impulsive  application  of  the  load  is 
to  be  expected,  an  appropriate  reduction  of  the  ordinary  working 
stress  should  be  made  to  provide  for  this  action.  Experience 
and  experiment  have  shown  that  the  repeated  variation  or  re- 
versal of  stress  affects  the  endurance  of  a  material,  sometimes 
causing  a  piece  to  break  under  a  load  which  it  has  often  pre- 
viously sustained.  The  theory  of  this  gradual  deterioration  is 
not  very  completely  developed  as  yet  ;  but  enough  has  been 
learned  to  show  that  the  working  stress  must  be  reduced  as  the 
magnitude  of  the  variations  of  stress  and  the  number  of  such 
variations  increases. 

The  quotient  of  the  ultimate  stress  divided  by  the  working 
stress  is  called  the  " factor  of  safety."  The  Table  [Unwin,  page 
24]  gives  some  general  values  of  the  factor  of  safety  for  a  few  of 
the  common  materials  with  constant  stress,  varying  stress  of 
one  kind,  reversal  of  stress,  and  shock.  Various  writeis  have 
given  such  tables,  and  a  comparison  of  the  factor  of  safety  recom- 
mended by  different  authorities  shows  a  very  wide  range.  See 
Thurston's  Text  Book  of  the  Materials  of  Construction,  page 
342  ;  Merriman's  Mechanics  of  Materials,  page  18,  find  many 
others.  All  such  general  values  should  be  looked  upon  simply 
as  suggestions  ;  for  the  proper  factor  of  safety  can  only  be  de- 
termined by  careful  study  of  the  conditions  of  the  particular  case 
in  hand.  It  is  frequently  proper  to  use  different  factors  of  safety 
for  different  members  of  the  same  structure  or  machine.  Differ- 
ent materials  and  the  methods  of  working  these  materials  make 
some  parts  more  liable  than  others  to  hidden  defects.  Certain 
members  may  be  subject  to  considerable  variation,  or  even  to  re- 
versal of  stress,  or  to  shock  ;  while  other  members  carry  a  load 
which  varies  much  less.  A  later  article  will  treat  more  fully  of 


,'     fig.  5. 


the  considerations  involved   in  determining  the  factor  of  safety 
appropriate  to  the  cases  which  ordinarily  arise. 

5.  Steady  or  Dead  Load,  and  Variable  or  Live  Load. — 
[Unwin.  $  18,  pages  24-25  ] 

6.  Stress  and  Strain  — [Unwin,  §§  18,  19,  pages  25-28.] 

7.  Resilience — [Unwin.  §  23,  page  38.]     If  a  material  is  dis- 
torted  by   a   straining  action,    it   is   capable   of  doing   a  certain 
amount  of  work  as  it  recovers  its  original  form.       If  the  deforma- 
tion does   not  exceed  the  elastic  strain,  this  amount  of  work  is 
equal  to  the  work  done  upon  the  material  in  producing  such  de- 
formation.    If  the  material  is  strained  beyond  the  elastic  limit, 
it  only  returns  work  equal  to  that  expended  in  producing  elastic 
deformation  ;  and   the  energy  required  to  cause  the  plastic  de- 
formation, or  set,  is  not  recovered,  as  it  is  not  stored   but  has 
been   expended   in    producing  such   permanent  change  of  form. 
Ordinary  springs  illustrates  the  first  case  ;  the  shaping  of  duc- 
tile  metals  by  forging,  rolling,  wire-drawing,  etc  ,  are  processes 
in  which  nearly  all  of  the  energy  is  expended  in  producing  set. 

The  work  required  to  produce  a  strain  in  a  member  is  called 
the  resilience.  If  the  strain  produced  is  equal  to  the  deformation 
at  the  true  elastic  limit,  the  energy  expended  is  called  elastic 
resilience*  If  the  piece  is  ruptured,  the  energy  expended  in 
breaking  it  is  called  ultimate  resilience  If  O ade  (Fig.  i)  is  the 
stress-strain  diagram  for  a  given  material  ;  the  area  O  a  a'  repre- 
sents the  elastic  resilience;  and  O  a  d e  e'  represents  the  ultimate 
resilience. 

In  such  materials  as  have  well  marked  elastic  limits  (propor- 
tionality between  stress  and  strain  through  a  definite  range)  the 
line  Oa  is  a  sensibly  straight  line,  and  the  elastic  resilience, 
Oaa'  —  \aa  X  Oa' ;  or,  the  clastic  resilience  equals  the  elastic  strain 
(Oa')  multiplied  by  one-half  the  elastic  stress  (^aa').  The  area 
Oadce  equals  the  base  (Oer)  multiplied  by  the  mean  ordinate  (y) 
of  the  curve  Oade  ;  or,  if  the  quotient  of  this  nit  an  ordinate  of 
the  curve  divided  by  the  maximum  ordinate  be  called  k,  the 

*  When  the  term  resilence  is  used  without  qualifying  context,  elastic  resi- 
lence  is  to  he  understood. 


—-6  — 

ultimate  resilience  equals  the  ultimate  strain  multiplied  by  k 
times  the  maximum  stress.  It  is  evident  that  for  a  straining 
action  beyond  the  elastic  limit,  £>  \  and  k  <  i. 

The  curve  OADEE'  represents  the  stress-strain  diagram  of  a 
material  having  higher  elastic  and  ultimate  strength  than  the 
former.  The  greater  inclination  of  the  elastic  line  (OA)  with 
the  axis  of  strain  (OX)  shows,  in  the  second  case,  a  higher 
modulus  of  elasticity,  as  this  modulus  equals  the  elastic  stress 

dd 
divided  by  the  elastic  strain.      In  the  first  case,  El  =  -^-f  ;    in  the 

AA' 

second  case,  E.^-  -=-— ,. 
U/l 

The  stress-strain  diagram  OADEE'  shows  that  of  two  materials 
one  may  have  both  the  higher  elastic  and  ultimate  strength,  and 
still  have  less  elastic  and  less  ultimate  resilience  If  the  curve 
O  a"  d" e"  is  the  stress-strain  diagram  of  a  third  material,  (having 
a  similar  modulus  of  elasticity  to  the  first)  it  appears  that  this 
third  material  possesses  greater  elastic  resilience,  but  less  ultimate 
resilience  than  the  first. 

A  comparison  of  these  illustrative  stress-strain  diagrams  (for 
quite  different  materials),  also  shows  that,  for  a  given  stress,  the 
more  ductile,  less  rigid  material  has  the  greater  resilience. 
Hence,  when  a  member  must  absorb  considerable  energy,  as  in 
case  of  severe  shock,  a  comparatively  weak  yielding  material 
may  be  safer  than  a  stronger,  stiffer  material.  This  is  frequently 
recognized  in  drawing  specifications  The  principle  is  the  same 
as  that  involved  in  the  use  of  springs  to  avoid  undue  stress  from 
shock.  In  fact  springs  differ  from  the  so-called  rigid  members 
only  in  the  degree  of  distortions  under  loads,  or  in  having  much 
greater  resilience  for  a  giv^n  maximum  load. 

If  a  material  is  strained  beyond  its  elastic  limit,  as  to  a  (  Fi^.  2), 
upon  removal  of  the  load  it  will  be  found  to  have  such  a  per- 
manent set  as  O  O '.  Upon  again  applying  load,  its  elastic  curve 
will  be  O1  a  \  but  beyond  the  point  a'  its  stress-strain  diagram 
will  fall  in  with  the  curve  which  would  have  been  produced  by 
continuing  the  first  test  (i.  e.,  a  de).  Similarly,  if  loaded  to  a", 
the  permanent  set  is  O  O",  and  upon  again  applying  load,  the 


-7- 

stress- strain  diagram  becomes  O"  a"  de.  The  elastic  limit  a"  of 
the  overstrained  material  is  evidently  higher  than  the  original 
elastic  limit,  a;  while  the  original  total  resilience,  Oade,  is 
considerably  greater  than  the  total  resilience  of  the  overstrained 
material,  O'  a"  de.  The  effects  of  strain  beyond  the  elastic  limit 
are  thus  seen  to  be  : 

I.  Elevation  of  the  elastic  strength  and  increase  of  the  elastic 
resilience. 

II.  Reduction  of  the  total  resilience. 

These  facts  have  an  important  influence  on  resistance  to  re- 
peated shock.  The  above  noted  elevation  of  the  elastic  limit  by 
overstraining  can  usually  be  largely  or  wholly  removed  by 
annealing. 

8.  Suddenly  applied  Load,  Impact,  Shock. — [Unwin,  §  18, 
pages  24-25  ;  §  21  a,  page  35.]  It  will  perhaps  be  well  to  first 
consider  the  general  case  of  a  load  impinging  on  the  member, 
with  an  initial  velocity  ;  this  velocity  (v)  corresponding  to  a  free 
fall  through  the  height  h.  For  simplicity,  the  discussion  will  be 
confined  to  a  load  producing  a  tensile  stress  ;  but  the  formulas  will 
apply  equally  well  to  compressive  and  uniform  shearing  stresses, 
and  all  except  (3)  and  (7)  apply  directly  to  cases  of  torsion  and 
flexure. 

W=  static  value  of  load  applied  to  member. 

//  =  height  corresponding  to  velocity  with  which  load  is  applied. 
/=  total  distortion  of  member  due  to  impulsive  load. 

P  =  maximum  intensity  of  resulting  stress. 
A  =  area  of  cross-section  of  the  member. 

P  =  p  A  =  total  max.  stress  due  to  load  as  applied. 

A  =  total  distortion  of  member  due  to  static  load,   W. 

x=h^\. 

k  =  a  constant ;    its  value  is  \  if  E.  L.   is  not  passed  ;  but  if 
E.  L.  is  exceeded  k  >  \  and  k  <  i. 

The  energy  to  be  absorbed  by  the  member  due  to  the  impulsive 
application  of  the  load  is  W(h  +  /)  ;  the  resilience  is  k  PI.  (See 


preceding  art.,  Resilience.)     The  expression   W(h  +  /)  gives  the 

W 
potential  energy  corresponding  to  the  kinetic  energy,        vl,  as 

given  by  Unvvin  on  §  23. 

Case  i . — Maximum  Stress  within  Elastic  Limit. 


W(h+l}=kPl=\Pl (i) 


P  =  PA=~^^L (3) 


l\\\\P\W     .-./=— (4) 

2lVh  2  Wh  2  W*h 

P=—r+2W=    px    +  2W=    px     +2W.     (5) 

W 


.  P*_*w*A 
x 

=  JT(i  +  ^/7T^).       .       .         (6) 

P  =  A  =  A  V  +  J1  +T"J=  A^l  +  V/I  +  2X^      ^ 


f  =£J-x(i  +  '/FT^)-      (8) 

"  jc=     !,/;=    X;       /^=    2.73   f-F;  /=  2.73  X. 
"  x=    4,  A=    4X;/>=     4  W7;         /=     4  X. 
"  ^=  I2,  A=  12  X;   />=     6  W^;         /=     6  X. 
"  x=  24,  //  =  24  X;  P=    8  IV-         /=     8  X. 
"  .*  =  4.0,  //  =  40  X;  T5^;  10  W;        /  =  10  X. 

As  X  is  small  for  metals  (except  in  the  forms  of  springs)  a  mod- 
erate impinging  velocity  may  produce  very  severe  stress.     It  will 


be  evident  that  X  a  nd  /  are  directly  proportional  to  the  length  of 
the  member  ;  hence  the  stress  produced  by  a  given  velocity  of 
impact  (height  Ji)  is  reduced  by  using  as  long  a  member  as  pos- 
sible. 

If  the  load  is  applied  instantaneously,  but  without  initial 
velocity,  h  =  o  and  x  --  o  ;  and  we  have  : 

/>=  W(  i  +  v/ 7+~o)  =  2  W.  (6') 

,----          .      .  ..     .     .     (/) 

/=X(i  +  v/f+^)  =  2X.         .        "'.         .         .  (8') 

Case  II. — Maximum  Stress  beyond  the  Elastic  Limit.  If 
the  maximum  stress  exceeds  the  elastic  limit,  the  constant  k 
of  eq.  (i)  is  between  ^  and  i,  (See  Art.  7,  Resilience),  and  its 
exact  value  cannot  be  determined  in  the  absence  of  the  stress- 
strain  diagram  for  the  particular  material.  Thus,  (Fig.  3)  W 
(h-\-l),  is  represented  by  the  rectangle  mncq\  and  this  area 
must  equal  the  resilience  area  Oahc;  the  latter  being  greater 
than  the  elastic  resilience,  O  a  a\  and  less  than  the  total  resili- 
ence, Oadee,  in  this  illustration. 

When  the  stress-strain  diagram  is  known,  the  following 
problems  can  be  readily  solved  : — 

(a)  Determination  of  the  velocity  of  impinging  of  a  given  load 
(or  corresponding  value  of  k}  to  produce  a  given  stress,  or  strain. 

(b)  Determination  of  the  load  which  will  produce  any  particu- 
lar stress,  ur  strain,  when  impinging  with  a  given  velocity. 

(c)  Determination  of  the  stress,  or  strain,  produced   by  a  given 
load  impinging  with  a  given  velocity. 

Let  the  resilience  corresponding  to  the  known  stress,  or  strain, 
in  (a)  and  (b),  be  called  R=  kPl.  If  the  stress-stmin  diagram 
is  for  stress  per  unit  of  sectional  area  and  strain  per  unit  of  length 
of  the  member,  let  W  be  the  load  per  unit  of  sectional  area,  and 
h'  be  the  height  due  the  velocity  of  impinging  divided  by  the 
total  acting  length  of  the  member. 


(a):      W(h'  +  l)  =  kPl=  R. 

•' 


(c)  :  The  solution  of  this  problem  is  not  quite  so  definite,  in 
the  general  case,  as  the  preceding  ;  but  it  can  be  easily  accom- 
plished, graphically,  with  sufficient  accuracy.  Draw  the  line 
g  q  (Fig.  3)  (indefinitely),  parallel  to  O  <?',  and  at  a  distance  from 
it  equal  to  W  ;  take  out  the  area  fig  ~g  Of.  Whatever  the 
value  of/,  the  shaded  area  OcqfigO=  W  I  ;  hence  the  un- 
shaded area  under  the  stress-strain  curve  must  equal  W  h'  .  A 
few  trials  will  suffice  to  locate  the  limiting  line  bqc  which  will 
give//  b  qf=  mn  O  t  =  W  h'. 

The  case  in  which  the  maximum  stress  is  within  the  elastic 
limit  is  by  far  the  most  important,  as  it  is  almost  always  de- 
sired to  keep  the  maximum  intensity  of  stress,  P-s-A,  within 
the  elastic  limit  ;  especially  as  every  overstrain  (beyond  this 
limit)  raises  the  elastic  limit  and  decreases  the  total  resilience 
[see  Fig.  2].  The  effect  of  a  shock  which  strains  a  member 
beyond  the  elastic  limit  is  to  reduce  its  margin  of  safety  for 
subsequent  similar  loads,  because  of  reduction  in  its  ultimate 
resilience.  Numerous  successive  reductions  of  the  total  resilience 
by  such  actions  may  finally  cause  the  member  to  break  under  a 
load  which  it  has  often  previously  sustained. 

No  doubt  many  cases  of  failure  can  be  accounted  for  by  the 
effects  just  discussed;  but  there  is  another  and  quite  different 
kind  of  deterioration  of  material,  which  is  treated  in  the  follow- 
ing article. 

Dr.  Thurston  has  shown  that  the  prolonged  application  of  a 
dead  load  may  produce  rupture,  in  time,  with  an  intensity  of 
stress  considerably  below  the  ordinary  static  ultimate  strength 
but  above  the  elastic  stress.  It  is  well  known  that  an  appreciable 
time  is  necessary  for  a  ductile  metal  to  flow,  as  it  does  flow  when 
its  section  is  changed  under  stress  ;  hence,  a  test  piece  will  show 
greater  apparent  strength  by  quickly  applying  the  load  than  by 


applying  it  more  slowly,  provided  the  application  of  load  is  not 
so  rapid  as  to  become  impulsive. 

The  kind  of  failure  which  is  the  subject  of  the  next  topic  is  due 
to  a  real  permanent  deterioration  of  the  metal,  and  it  is  due  to 
distinctly  different  causes  than  those  mentioned  above. 

9.  On  the  Peculiar  Action  of  Live  Load.  Fatigue  of  Metals. 
— [Unwin,  §21,  pjges  29—34.]  Exception  may  be  taken  to  the 
statement  (fifth  line  from  bottom  of  page  29)  :  "  Here  the  factor 
2  or  3  is  a  real  factor  of  safety  which  allows  for  unknown  contin- 
gencies." A  factor  of  nearly  2  is  required  to  allow  for  the 
known  difference  between  the  ultimate  and  the  elastic  strengths  ; 
so  that  when  the  working  stress  is  ^  the  ultimate  stress,  there  is 
scarcely  any  margin  for  contingencies  ;  and  with  a  working  stress 
of  YZ  the  ultimate  stress  the  factor  for  contingencies,  above  the 
point  at  which  permanent  set  is  to  be  expected,  is  only  about 

3--2=-I^. 

It  has  been  found  by  experience  and  experiment,  that  materials 
which  are  subjected  to  continuous  variation  of  load  cannot  be  de- 
pended upon  to  resist  as  great  stress  as  they  will  carry  if  applied 
but  once,  or  only  a  few  times.  When  the  load  is  suddenly  ap- 
plied, and  frequently  repeated,  the  decline  of  strength  or  of  the 
power  of  endurance,  may  perhaps  be  ascribed,  in  part  at  least,  to 
the  elevation  of  the  elastic  limit  and  reduction  of  the  ultimate 
resilience,  as  discussed  in  Art.  8.  But  apart  from  this  cause, 
with  repeated  loads,  even  in  the  absence  of  appreciable  shock, 
a  decided  deterioration  of  the  material  very  frequently  occurs. 
This  effect  has  been  called  the  Fatigue  of  Materials,  although 
Unwin  (§  2ia)  restricts  this  term  to  the  kind  of  deterioration 
already  referred  to  as  the  simple  result  of  a  decrease  of  resilience. 
The  term  fatigue  implies  a  weakening  of  the  material  due  to  a 
general  change  of  structure.  It  was  formerly  commonly  sup- 
posed that  the  repeated  variation  of  stress  caused  such  change  of 
structure,  possibly  owing  to  slight  departure  from  perfect  elas- 
ticity under  stress  much  below  that  ordinarily  designated  as  the 
elastic  limit.  The  crystaline  appearance  of  the  fracture  sustained 
this  view  ;  but  numerous  tests  of  pieces  from  a  member  ruptured 
in  this  way,  (taken  as  near  as  possible  to  the  break),  fail  to  show 


such  crystaline  fracture,  and  it  is  difficult  to  reconcile  the  normal 
appearance  and  behavior  of  such  test  pieces  with  the  theory  of 
general  change  of  structure. 

Every  piece  of  metal  contains  innumerable  minute  flaws 
or  imperfections,  often  originally  too  small  to  be  detected 
by  ordinary  means.  These  "micro-flaws"  tend  to  extend 
across  the  section  under  variation  of  stress,  and  mny,  in  time, 
reduce  the  net  sound  section  so  greatly  that  the  intensity  of 
stress  in  the  fibres  which  remain  intact  becomes  equal  to 
the  normal  breaking  strength  of  the  material.  Professor 
Johnson  suggests:  "the  gradual  fracture  of  metals"  as  a 
more  appropriate  term  than  "fatigue."  Many  men  of  large 
practical  experience  still  prefer  wrought  iron  to  mild  steel  for 
various  members  which  are  subject  to  constantly  reversing  stress. 

It  is  probable  that  the  prejudice  against  steel  is  largely  the  re- 
sult of  unskillful  manipulation  of  this  more  sensitive  material  ; 
and  the  product  of  the  best  steel  makers  of  today  is  much 
stronger  and  more  reliable  than  wrought  iron. 

However,  the  very  lack  of  homogeneity  in  wrought  iron  renders 
it  safer  under  varying  stress,  (other  things  being  equal ),  as  the 
fibres  are  more  or  less  separated  by  the  streaks  of  slag,  and  a  flaw 
is  less  apt  to  extend  across  the  entire  section  than  it  is  in  the  con- 
tinuous structure  of  steel.  Wrought  iron  ma}r  be  likened  to  a 
wire  rope,  in  which  a  fracture  in  one  wire  does  not  directly  ex- 
tend to  adjacent  wires. 

The  "gradual  fracture"  through  extension  of  "  micro  flaws  " 
seems  to  accord  with  the  observed  facts  more  closely  than  the 
older  theory  of  general  change  of  structure. 

The  theory  of  the  subject  is,  as  yet,  too  incomplete  to  permit 
of  derivation  of  rational  formulas  to  account  for  the  effects  of  re- 
peated live  loads  ;  and  if  the  "  micro-flaw  "  theory  is  correct,  it 
is  not  probable  that  such  rational  analysis  can  ever  be  satis- 
factorily applied. 

All  of  the  formulas  that  have  been  derived  for  computation  of 
breaking  strength  under  known  variations  of  load,  or  stress,  are 
empirical  ones  which  have  been  adjusted  to  fit  the  experiment- 
ally determined  facts. 


-13  — 

Consult :  Johnson's  Materials  of  Construction. 
Merriman's  Mechanics  of  Materials. 
Unwin's  Testing  of  Materials. 
Weyrauch  (Du  Bois)  Structure  of  Iron  and  Steel. 

Experiment  has  shown  that  the  breaking  strength  under  re- 
peated loading,  or  the  "  carrying  strength  ",  is  a  function  of  the 
magnitude  of  the  variation  of  stress  and  of  the  number  of  repeti- 
tions of  such  varying  stress.  Furthermore,  this  function  is  dif 
ferent  for  different  materials  ;  and  there  are  authentic  observa- 
tions on  record  which  go  to  show  that,  as  between  different 
materials,  the  one  with  the  higher  static  breaking  strength  does 
not  always  possess  the  greater  endurance  under  repeated  loading. 
In  general,  however,  the  carrying  strength  under  repeated  loads 
is  a  function  of  the  static  strength. 

The  allowable  working  stress  u>ually  depends  upon  :  (a),  The 
number  of  applications.  This  should  be  considered  as  indefinite, 
or  practically  infinite,  in  many  machine  members,  (b),  The 
range  of  load.  This  is  frequently  either  from  zero  to  a  maximum  ; 
or  between  equal  plus  and  minus  values,  (c),  The  static  break- 
ing strength  or  the  elastic  strength. 

The  first  systematic  experiments  upon  the  effect  of  repeated 
loading  were  conducted  by  Wohler  [1859  to  1870].  He  found, 
for  example,  that  a  bar  of  wrought  iron,  subjected  to  tensile 
stress  varying  from  zero  to  the  maximum,  was  ruptured  by  : 

800  repetitions  from  o  to  52,800  Ibs.  per  sq  in. 
107,000         "  "     o  "  48,000 

450,000         "  "     o  "  39,000         "         " 

10,140,000  '     o  "  35,000 

[Merriman,  page  191]. 

It  was  found  that  the  stress  could  be  varied  from  zero  up  to 
something  less  than  the  elastic  limit  an  indefinite  number  of 
times  (several  millions)  before  rupture  occurred  ;  but  with  com- 
plete reversal  of  stress,  or  alternate  equal  and  opposite  stresses, 
(tension  and  compression),  it  could  be  broken,  by  a  sufficient 
number  of  applications,  when  the  maximum  stress  was  only  about 
one-half  to  two-thirds  the  stress  due  to  the  elastic  limit. 


-  14  — 

The  general  formula  given  by  Unwin,  (eq.  (i),  page  32),  for 
the  maximum  carrying  strength,  is  : 

A»«=  ~  +  <S~K*-n±K 

This  expression  can  be  used  when  the  range  of  intensity  of 
stress,  ±,  (i.e.,  the  difference  between  minimum  and  maximum 
unit  stress)  is  known.  It  is  not  directly  applicable,  however,  to 
the  general  problem  of  design,  which  is  to  determine  the  dimen- 
sions of  a  member  to  safely  carry  a  load  varying  between  given 
limits  ;  for  the  variation  of  unit  stress  (intensity  of  stress)  due  to 
the  range  of  load  is  not  known  until  these  dimensions  are  known. 
Thus,  if  the  load  on  a  wi  ought  iron  tension  member  varies  be- 
tween 3  tons  and  20  tons  an  indefinite  number  of  times,  it  is  de- 
sired to  determine  the  proper  area  of  cross-section  ;  but  the  value 
of  A  cannot  be  assigned  until  this  section  is  known.  Three 
special  forms  of  the  above  expression  are  given  (Unwin,  page  32) 
for  the  special  cases  of:  (i)  dead  load,  £max=£m)n,  A  =  o; 
(2)  repeated  load  (stress  entirely  removed  but  never  changing 
sign),  £min  =  o,  A---£max;  and  (3)  complete  reversal  of  load 
(equal  and  opposite  stresses  alternating),  k  mln  =  —  k  max, 
A  =  2^max.  These  special  formulas  are  generally  applicable  to 
the  appropriate  cases,  because  A  has  been  eliminated.  However, 
it  appears  best  to  adopt  a  formula  given  by  Professor  Johnson, 
as  it  is  applicable  to  all  cases  that  will  arise  ;  it  is  .simpler  than 
most  of  those  previously  proposed  ;  and  it  is  probably  as  reliable 
as  any  yet  offered. 

Two  formulas  which  have  been  very  generally  accepted  for 
computing  the  probable  carrying  strength  are  :  Launhardt's  for 
varying  stress  of  one  kind  only,  and  Weyrauch's  for  stress  which 
changes  sign. 

Suppose  a  material  to  have  a  static  ultimate  strength  of 
60,000  Ibs.  per  sq.  in.  If  the  minimum  unit  strength  be  plotted 
as  a  straight  line,  —p^  Of,  (Fig.  4),  the  locus  of  the  maximum 
unit  stress,  from  the  Launhardt  formula,  is  the  broken  curve  from 
d\.Q  f.  That  is,  for  example,  when  the  minimum  tensile  stress 
is  15,000,  the  maximum  tensile  carrying  stress  would  be  about 


-15- 

40,000  ;  or  the  material  could  be  expected  to  stand  an  indefinite 
number  of  loadings  if  the  range  of  stress  did  not  exceed  15,000  to 
40,000  pounds  per  square  inch  tension.  In  a  similar  way,  the 
broken  curve  from  c  to  d  is  the  locus  of  maximum  tension,  from 
the  Weyrauch  formula,  when  the  locus  of  minimum  stress 
(negative  tension,  or  compression)  is  the  straight  line  —  p.2  O.  It 
will  appear  that  the  straight  line  p^p^f  agrees  fairly  well  with 
these  two  curves.  Inasmuch  as  it  seems  unreasonable  to  expect 
an  abrupt  change  of  law  when  the  minimum  stress  passes 
through  zero,  and  as  there  is  no  rational  basis  for  the  Launhardt 
and  Weyrauch  formulas,  it  appears  reasonable  to  adopt  the 
upper  straight  line  as  the  locus  of  the  maximum  stress.  Owing 
to  the  discrepancies  in  the  observations  (which  must  be  expected 
from  the  probable  cause  of  the  deterioration  of  the  metal),  this 
straight  line  may  be  accepted  as  representing  the  law  as  accu- 
rately as  could  be  expected  of  any  empirical  line.  These  are,  in 
substance,  the  reasons  given  by  Professor  Johnson  for  basing  his 
formula  on  the  straight  line  p^p^f.  For  full  discussion  and  deri- 
vation of  the  following  formula,  see  Johnson's  Materials  of 
Construction,  pages  545-547. 

Let  p  =  maximum  intensity  of  stress. 
P'  =  minimum  intensity  of  stress. 

u  —  ultimate  (static)  intensity  of  stress. 

u'  —  ultimate  (static)   intensity  of    stress   divided   by  the 
proper  factor  of  safety. 

f=  working  stress. 
Then  : 


<" 


As  the  expressions  contain  the  ratio  of  the  minimum  to  maxi- 
mum intensities  of  stress,  instead  of  their  difference,  they  are 
applicable  when  the  area  of  cross-section  of  the  member  is  un- 


—  16  — 

known  ;  for  whatever  this  area,  the  ratio  of  the  stresses  is  the 
same  as  the  ratio  of  the  loads  producing  these  stresses.  In  sub- 
stituting values  of  p  and  p'  ,  care  must  be  taken  to  use  proper 
signs  ;  thus,  if  tension  is  taken  as  positive,  compression  is 
negative  ;  or,  if  the  stress  varies  between  tension  and  compression, 
p  is  positive  and  p'  is  negative. 

For  dead  load,    >'=; 


r  U  U  ,  ,      ,  N 

•••/=-     -r=  \-=n'  .....       (2') 

'     * 

For  repeated  load  when  p'  =  o,  or  2-  =  o 

P 


•••/=r~o=^' 

For  complete  reversal  of  load,  p'  =  —p, 


10.  The  Real  Factor  of  Safety.  —  [Unwin,  page  34.  Deter- 
mination of  Safe  Working  Sttcss.~\  As  shown  in  the  preceding 
article,  the  safe  working  stress  for  a  given  material,  when  sub- 
jected to  repeated  variation  of  load,  should  be  less  than  that 
which  could  be  safely  allowed  for  the  same  material  under  a  dead 
load. 


Machine  members  are  usually  subjected  to  varying  stresses  ; 
the  load  is  frequently  applied  so  rapidly  as  to  constitute  an  im- 
pulsive action  ;  and  the  kinetic  conditions  often  introduce  con- 
siderable additional  stress  of  such  complex  nature  as  to  preclude 
very  exact  analysis.  Owing  to  these  elements,  a  lower  working 
stress  is  necessar)-  in  many  machine  members  than  is  required,  for 
equal  safety  with  the  same  material,  in  stationary  structures  sub- 
jected to  a  nearly  constant  load. 

The  apparent  factor  of  safety,  which  is  the  quotient  of  the  static 
ultimate  strength  divided  by  the  working  stress,  is  often  8,  10, 
12.  or  more,  in  machine  parts  ;  while  it  may  be  only  3  or  4  in  a 
structure  with  a  nearly  stead}'  load.  But  it  does  not  necessarily 
follow  that  \\\o.  real  factor  of  safety,  or  margin  allowed  for  such 
contingencies  as  defects  of  material  and  workmanship  is  any 
larger  in  the  former  than  in  the  latter  case. 

In  fact  the  usual  margin  for  such  contingencies  is  not  larger 
in  machine  members,  or  there  would  be  correspondingly  fewer 
failures  of  these  parts  in  regular  service.  It  is  true  that  the  ele- 
ments of  uncertainty  in  the  total  straining  action  on  a  machine 
are  often  more  numerous,  and  of  greater  magnitude  relative  to 
the  primary  straining  action  due  to  the  "useful  load"  ;  hence 
the  total  contingency  factor  may  proper!}'  be  greater  than  in  a 
bridge  or  roof  truss. 

The  factor  of  safety  has  been  called  the  "  factor  of  ignorance," 
and  as  it  is  too  often  applied  it  is  perhaps  little  else.  It  is  proba- 
ble that  the  factor  of  safety  must  always  retain  an  element  of 
ignorance ;  for  it  can  hardly  be  hoped  that  the  powers  of 
analysis  will  ever  permit  the  prediction  of  the  exact  effect  of  every 
possible  straining  action,  due  to  regular  service  and  accident ; 
neither  can  it  be  expected  that  the  methods  of  manufacture  and 
of  inspection  will  become  so  perfect  as  to  eliminate,  or  measure 
precisely,  every  possible  defect  in  materials  and  workmanship. 

The  development  of  the  theory  of  actions  which  are  as  yet  but 
imperfectly  understood,  may  reduce  the  element  of  ignorance. 
However,  a  careful  study  of  the  conditions  of  each  particular 
case,  and  proper  attention  to  effects  which  may  be  weighed  (at 
least  approximately)  in  the  present  state  of  knowledge  should 


—  18  — 

lead  to  a  much  more  intelligent   employment   of  the   factor  of 
safety  than  is  common  today. 

It  has  been  said  that:  "The  capacity  to  decide  upon  the 
proper  factor  of  safety  is  the  important  point  in  design."  It  is 
certainly  not  reasonable  to  make  long,  tedious  computations,  the 
results  of  which  depend  upon  a  carelessly  chosen  factor  of  safety. 
One  who  cannot  determine  a  rational  factor  of  safety,  can  derive 
little  benefit  from  the  use  of  a  rational  formula.  In  short,  the 
selection  of  the  proper  constants  is  the  part  of  the  engineer  ;  the 
computation  is  the  part  of  a  clerk. 

Most  of  the  formulas  of  mechanics,  as  applied  to  questions  of 
strength  in  design,  are  based  upon  theoretical  treatment  of  the 
stresses  induced  by  the  action  of  given  forces  on  the  parts  under 
consideration.  There  are  many  cases  in  which  this  course  is  per- 
fectly logical,  and  the  conclusions  are  irresistible  ;  while,  in  many 
other  instances,  members  of  a  machine  or  structure  are  subjected 
to  such  a  complicated  system  of  stresses  that  analysis  cannot  be 
strictly  applied,  and  less  satisfactory  approximations,  or  assump- 
tions, are  unavoidable,  in  the  present  state  of  knowledge.  This 
last  condition  of  things,  which  is  not  unusual  in  the  design  of 
machines,  introduces  the  first  of  many  elements  of  uncertainty, 
and  one  of  three  methods  of  arriving  at  the  proportions  of  the 
parts  is  possible.  First,  if  the  predominating  action  is  capable  of 
rational  treatment,  the  member  can  be  designed  as  if  for  the  corre- 
sponding stresses,  and  such  a  margin  as  is  dictated  by  experience 
or  experiment  may  then  be  allowed  for  the  more  uncertain  ele- 
ments. Second,  analysis  may  be  abandoned  and  resort  may  be 
had  to  empirical  formulas  derived  from  experiment.  Third,  the 
last,  and  not  most  uncommon,  recourse  is  to  "judgment."  This 
last  method,  when  it  is  real  judgment,  based  upon  a  large  experi- 
ence, has  produced  magnificent  results ;  in  many  cases, 
(especially  for  details  and  small  parts),  it  is  the  only  way  to 
proceed. 

The  general  nature  of  the  factor  of  safety,  and  the  effects  of 
shock  and  of  repeated  stresses  have  been  discussed  in  preceding 
articles. 

If  the  working  stress  due  to  the  total  regular  straining  action, 


—  19  — 


and  the  stress  which  the  material  will  sustain  indefinitely  (under 
the  conditions  to  which  it  is  subjected)  are  known,  it  is  only 
necessary  to  so  proportion  the  members  that  the  latter  stress  will 
exceed  the  former  by  margin  enough  to  cover  such  contingencies 
as  over  loading  and  defects  of  material  and  of  workmanship  as 
might  reasonably  be  expected  to  possibly  occur. 

The  following  comparisons  give  an  idea  of  what  this  con- 
tingency margin,  or  real  factor  of  safety  is,  with  such  working 
stresses  as  are  not  uncommonly  allowed  in  machines.  The 
Static  Breaking  Strengths  of  the  various  materials  are  assumed 
as  fair  representative  values.  Each  Working  Stress  is  obtained 
by  dividing  the  static  breaking  strength  by  the  appropriate  ap- 
parent factor  of  safety  as  given  in  the  table  on  page  24  of  Un win's 
Machine  Design.  The  Carrying  Strength  is  taken  as  follows  : 

Case  I.    Dead  Load  ; — Carrying  strength  =  static  strength. 

Case  II.  Repeated  Load, — applied  and  removed  an  indefinite 
number  of  times  but  stress  of  one  kind  only  (tension)  ; — Carrying 
strength  =  %  static  strength. 

Case  III.  Reversal  of  Load, — stress  varying  an  indefinite 
number  of  times  between  equal  tension  and  compression, — 
Carrying  strength  =  y$  static  strength. 

The  Real  Factors  of  Safety  are  obtained  by  dividing  the 
Carrying  Strengths  by  the  corresponding  working  stresses. 


CAST  IRON. 

WRO'T  IRON. 

MACH'Y  STEEL. 

» 

Carrying  Strength  
Working  Stress 

17,000 
4,250 

56,000 
18,670 

65,000 
21,670 

0 

Real  Factor  of  Safety  _. 

4 

3 

3 

M 

Carrying  Strength  
Working  Stress 

8,500 
2,830 

28,000 
II  200 

32,500 
13  ooo 

cJ 

Real  Factor  of  Safety  __ 

3 

2^ 

2% 

M 

Carrying  Strength  
Working  Stress  _ 

5,670 
1,700 

18,670 
7,OOO 

21,670 
8,125 

O 

Real  Factor  of  Safety  ._ 

3l/3 

2% 

2% 

It  will  appear  from  the  above  table  that  the  real  factors  of 
safety  are  rather  less  for  Cases  II  and  III  than  those  taken  for  a 
dead  load  (Case  I)  ;  hence  the  high  apparent  factors  do  not  pro- 
vide an  excessive  margin  for  the  contingencies  likely  to  occur. 
In  fact  this  margin  is  comparatively  small  ;  for  the  liability  to 
extra  straining  action  is  greater  with  live  load  than  with  dead 
load. 

If  there  is  apt  to  be  much  shock,  the  resulting  stresses  may  be 
much  beyond  those  due  to  the  gradual  application  of  the  load,  as 
shown  in  article  8  ;  and  it  will  be  evident  that  the  apparent  fac- 
tors of  safety  of  15  and  12,  given  by  Unwin  for  cast  iron  and 
wrought  iron  or  steel,  respectively,  are  not  excessive  under  such 
conditions.  In  some  instances,  shock  is  so  violent  and  indeter- 
minate that  the  necessary  factor  of  safety  becomes  so  large  as  to 
render  computations  of  very  little  value  in  proportioning  mem- 
bers, and  the  practical  machines  of  this  class  are  the  products  of 
a  process  of  evolution. 

The  importance  of  a  knowledge,  upon  the  part  of  the  designer, 
of  the  methods  emplo3?ed  in  the  manufacture  of  the  materials 
used,  and  of  the  practice  of  the  shops  in  which  the  designs  are 
executed,  is  appreciated  when  it  is  considered  that  these  things 
all  have  a  direct  influence  upon  the  proper  factor  of  safety.  As 
improved  methods  of  the  metallurgist  insure  a  more  reliable  and 
homogeneous  product,  and  as  methods  of  inspection  are  perfected, 
the  danger  from  hidden  defects  in  the  material  furnished  becomes 
less  ;  as  artisans  become  more  accustomed  to  the  properties  of 
the  material  which  they  handle,  and  learn  to  respect  its  weak- 
nesses ;  as  investigators  develop  the  effect  of  repeated  stresses  of 
the  various  kinds  ;  and  as  the  engineer  learns  to  study  all  of  these 
elements  and  to  give  to  each  its  due  weight ;  the  factor  of  safety 
will  be  reduced,  it  will  become  less  and  less  a  factor  of  ignorance, 
and  more  and  more  a  true  factor  of  safety 

n.  Straining  Action  due  to  Power  transmitted.— [Unwin, 
§  22,  page  36.] 

The  expression,  P=-*$-HP,  may  be  used  to  find  the  strain- 


ing  force  in  a  chain  transmitting  power  between  sprocket  wheels  ; 
provided,  (as  is  usual),  the  chain  is  so  loose  that  only  the  driving 
side  is  under  any  considerable  tension.  With  endless  belt  or  rope 
transmission,  where  the  friction  between  the  band  and  the  wheels 
is  depended  upon  to  prevent  undue  slipping,  it  is  necessary  to 
have  considerable  tension  on  the  "  idack  "  side  to  secure  sufficient 
adhesion.  In  such  cases,  the  above  expression  gives  the  effective 
pull  due  to  the  power  transmitted,  or  the  difference  between  the 
total  tensions  on  the  two  straight  portions  of  the  band.  The 
maximum  straining  force  on  the  tight,  or  driving,  side  is  equal  to 
P  (as  given  above)  plus  the  pull  on  the  slack  side  ;  this  total  ten- 
sion on  the  driving  side  is  frequently  twice  P,  or  even  more. 

It  is  often  convenient  to  use  the  velocity  in  feet  per  minute  (  F), 
in  place  of  the  velocity  in  feet  per  second  (v)  of  the  transmitting 
connector  (link,  rod  or  band).  V  —  60  v, 


V 

Whatever  the  path  of  the  point  of  application  of  the  load,  the 
resultant  force  acting  upon  this  point  (due  to  the  load  combined 
with  the  constraining  forces)  must  lie  along  the  tangent  to  the 
path.  [Newton's  Laws.] 

When  the  path  of  the  moving  connector  does  not  coincide  with 
the  line  of  its  axis,  the  total  straining  force  along  this  axis  can  be 
found  by  multiplying  /'(computed  as  above)  by  the  secant  of  the 
angle  which  the  direction  of  the  load  force  makes  with  the  direc- 
tion of  the  axis  of  the  connector. 

In  Figs.  5  and  6,  let  :  — 

P-—  useful  load  force  applied  at  C, 

PI  =  corresponding  resistance  overcome  at  c, 

P'  =  resultant  force  at  C,  along  axis  C  .  .  c, 

PI  =  resultant  force  at  c,  along  axis  C  .  .  c, 

P"  =  constraining  force  at  C, 

P"  =  constraining  force  at  c. 

a!  =  90°  -  a  ;   &  =  90°  -  /3. 

Pf  =  P  sec  a'  =  P  cosec  a  =  P  -4-  sin  a.  (2) 

/Y  =  P,  sec  P  =  P!  cosec  ft  =  P,  -=-  sin  ft.  (3) 


—  22  — 

It  is  evident  that  />/  is  equal  and  opposite  to  P  ;  or  P}'=  —  P', 
if  the  connector  (Cc)  is  moving  with  uniform  velocity  ;  for  con- 
sidering the  connector  as  a  free  body  acted  upon  by  these  two 
opposite  resultant  forces,  inequality  of  these  forces  would  produce 
acceleration  (positive  or  negative). 

The  resistance  (/\)  overcome  at  c  is  only  equal  to  the  driving 
force  (P)  acting  at  C,  when  these  two  points  have  equal  velocities  ; 
for  the  energy  applied  at  C  equals  the  energy  delivered  at  c  (ne- 
glecting losses  due  to  friction)  ;  hence  the  forces  acting  at  C  and 
c  are  inversely  as  the  velocities  of  these  points.  The  resistance 
(/*,)  corresponding  to  a  known  driving  force  (P)  can  be  found  as 
follows  : 

From  the  necessary  conditions  for  equilibrium,  2,  moms  =  o  : 

P.OC  =  P,-Oc 
O  C  \  O  c  ::  sin  ft  :  sin  a 
.'.  Ps'm  (3=  PI  sin  a.  (4) 


The  expression  M=  55°  ^corresponds  with  />=  55°-^; 
2  TT  n  v 

for  if  r  =  the  radius  at  which  the  force  P  acts,  v  =  2  tr  r  n 


...  P==  55Q  /^    .    Pr=M=  55Q  HP 
2  TT  r  n  2  TT  n 

If  r"  is  in  inches  and  P  in  pounds,   M  is  the  moment  in  inch- 
pounds. 

ff  P 

The   expression:      J/  =63024  —  ---  should    be    committed    to 

memory. 

12.  Straining   Actions    due   to    Variations  of  Velocity.  — 
[Unwin,  §22,  pages  36-37.] 

If  the  acceleration,      v,  be  represented  by±/»,  the  force  required 
at 

to  produce  this  acceleration  is  ±  —    ~  =  =fc  —  />,  and  the  stress 

g  dt  £ 

produced  in  a  member  which  transmits  this  force  is  q=  —  /. 

g 


-23- 

Umvin   refers  to  p  (page  37)  as  the  acceleration    "  per  unit  of 

W 
weight",  and  he  calls — p  the   "total  acceleration."     It  is  more 

o 

exact  to  consider^*  as  the  acceleration  (rate  of  change  of  velocity), 
which  is  numerically  equal  to  the  accelerating  force  per  unit  of 

mass,  and       p  as  the  total  accelerating  force . 

<*> 

Referring  to  Unwin's  Fig.  6.  let  the  angle  which  the  tangent  at 
b  makes  with  the  axis,  Ox,  be  called  a.  Then  the  velocity 
(v,  =^a  b)  is  increasing  at  a  rate  which  is  proportional  to  sin  a  ; 
and  the  distance  (s,  =  Oa),  or  space  passed  over,  is  increasing  at 
a  rate  proportional  to  cos  a.  If  the  unit  of  velocity  and  unit  of 
space  passed  over  are  plotted  to  the  same  scale,  a  d  represents  the 
acceleration  to  a  similar  scale,  for  : 

d  v  '.  d  s  ','.  sin  a  :  cos  a  ;  also, 
a  d  :  a  b  ::  sin  a  :  cos  a, 
. '.  d  v  '.  d  s  '.'.  ad:  a  b 

dv    ds 

.'.-=—'»    ,-  '.'.  a  a  '.  a  o 
d  t    dt 

But.   r-  =  the  acceleration,  and 
a  t 

=  the  velocity;    therefore  if  a£=the  velocity  to  any 

scale,  ad—  the  acceleration  to  a  corresponding  scale. 

The  accelerating  force  (F)  equals  the  mass  multiplied  by  the 

W          W 
acceleration,    or     F  =        p=     -Xad.     In  a  velocity  diagram 

on  a  space  base,  (z.  <?.,  with  abscissas  representing  space  traversed, 
and  ordinates  representing  corresponding  velocities)  the  sub- 
normal (a  d)  gives  the  acceleration.  In  velocity  diagrams  plotted 
on  a  time  base  (z.  e.,  abscissas  representing  time  instead  of  dis- 
tance) the  subnormal  does  not  give  the  acceleration. 


—  24  — 

13.  Stress  due  to  Change  of  Temperature.  [Unwin,  §  23, 
page  38]. 

If  a  bar  is  subjected  to  a  change  of  temperature,  of  /  degrees, 
it  would,  if  free  to  expand,  increase  in  length  from  /to  /(i  -fa/); 
a  being  the  coefficient  of  expansion.  If  expansion  is  prevented 
by  a  constraining  force,  this  force  equals  that  required  to  shorten 
the  bar  by  the  amount  la.  t,  or  to  shorten  each  unit  of  length  by 
at.  Hence  the  intensity  of  stress,  /,  due  to  prevention  of  ex- 
pansion is  that  corresponding  to  the  unit  strain  at  Since  the 
modulus  of  elasticity,  E,  equals  the  stress  divided  by  the  corre- 
sponding strain  (within  the  elastic  limit),  E=  .-'•/  =  Ea  t. 

The  following  values  may  be  taken   for  the  coefficient  of  ex- 
pansion per  one  degree  Fahrenheit, 
Cast  Iron  —  .0000062. 

Wrought  Iron     —  .0000068. 
Soft  Steel  =  .0000060. 

Hardened  Steel  =  .0000070. 
Brass  =  .0000105. 

Bronze  =  .0000106. 


II. 

COMPOUND  STRESS. 


14.  Resistance  of  Columns,  or  Long  Struts. — [Unwin,  §§ 
37~39.  pages  77-81  ;  omitting  Grashof  formulas.] 

Very  short  compression  members,  of  ductile  material,  fail 
under  stress  corresponding  to,  or  only  slightly  in  excess  of,  the 
apparent  elastic  limit,  or  yield  point ;  for  when  this  stress  is 
reached  the  metal  flows,  although  it  does  not  actually  break. 
Very  long  columns  may  approximate  the  resistance  as  given  by 
Euler's  formulas  Columns  of  lengths  intermediate  between 
compression  members  which  yield  by  simple  crushing  and  those 
which  fail  by  pure  flexure  are  weaker  than  the  former  and 
stronger  than  the  latter.  If  a  column  is  initially  exactly  straight, 
perfectly  homogeneous,  and  subjected  to  an  absolutely  concentric 
load  (that  is,  if  it  is  an  ideal  column)  there  seems  to  be  no  reason 
why  its  strength'  should  diminish  rapidly  with  an  increase  of 
length,  other  conditions  remaining  the  same. 

However,  even  an  ideal  very  long  column  would  reach  the  con- 
dition of  unstable  equilibrium  when  subjected  to  a  certain  critical 
load  (the  greatest  load  consistent  with  stability).  If  the  load  is 
increased  beyond  this  limit  and  a  deflection  is  caused  in  any  way, 
the  deflection  will  increase  until  the  stress  due  to  flexure  pro- 
duces failure  of  the  column.  If  a  deflection  is  caused  while  the 
column  is  under  a  load  less  than  this  greatest  load  consistent 
with  stability,  the  elasticity  of  the  material  tends  to  make  the 
column  regain  its  normal  form.  Initial  defects  in  the  form  or 
structure  of  a  column  or  eccentric  application  of  load  tend  to  pro- 
duce such  a  deflection  ;  hence  long  struts  fail  under  smaller  loads 
than  short  struts  of  similar  material  and  cross-section,  for  ideal 
conditions  are  not  realized  in  practice  ;  or  for  equal  safety  under 


—  26  — 

a  given  load  long  columns  must  have  a  greater  cross-section, 
and  lower  mean,  or  nominal,  working  stress.*  Even  in  columns 
of  moderate  length,  if  of  ductile  material,  the  flow  at  the  yield 
point  causes  buckling. 

Merriman  says  that  if  the  length  of  a  compression  member  be 
only  from  four  to  six  times  its  least  "  diameter,"  it  may  be  treated 
as  one  which  will  yield  by  simple  compression.  Unwin  (page  81) 
gives  limits  within  which  the  Euler  formula  should  not  be  ap- 
plied. Other  authorities  give  somewhat  different  limits  ;  but 
nearly  all  agree  that  most  of  the  columns  in  ordinary  structures, 
and  machines  are  intermediate  between  simple  compression  mem- 
bers and  those  to  which  Euler's  formulas  apply.  The  limits 
assigned  by  Professor  Johnson  for  "  square  ended  "  columns  agree 
substantially  with  those  given  for  wrought  iron  by  Unwin.  There 
have  been  a  great  many  column  formulas  proposed.  A  graphical 
representation  of  several  of  these  formulas  is  shown  in  Fig.  7.  In 
this  diagram,  abscissas  represent  ratios  of  the  length  of  column  to 
the  least  radius  of  gyration  of  the  cross-section,  and  the  ordinates- 
represent  the  nominal  (mean)  intensity  of  compressive  stress.  Or, 

x  =/-i-/3=/-T-  v' 1 -=~  A,  and  ^  =  p  =  P  -j-  A,  when 

/    =  the  length  of  the  column, 

p   =  the  least  radius  of  gyration  of  cross-section, 

/  =  the  least  moment  of  inertia  of  cross-section, 

A  =  the  area  of  the  cross-section. 

P  =  the  breaking  load  on  the  column,     . 

P  —  the  mean  intensity    of  stress    under    the    breaking  load, 

or  the  unit  breaking  load,  --=  P-r-  A. 

The  following  additional  notation  is  also  used  in  this  article  : 
n  =  the  factor  of  safety, 


*Owing  to  the  flexure  of  the  long  column,  the  stress  is  not  uniform 
across  the  section.  The  maximum  intensity  of  stress  must  be  kept  within 
the  compressive  strength  of  the  material  ;  hence  the  mean  stress  is  less 
than  for  shorter  compression  members,  in  which  the  mean  stress  is  more 
nearly  equal  to  the  maximum. 


[+S 

100  200 

Fig.  7. 


2=^ 


V 


0        10      20  40  60  80  100  120 


-27- 

P'=  the  ivorking  load  on  the  column,  =  P '-5-  «, 

/>'  =  the  mean  intensity  of  working  stress,  or  unit  working  load, 
^P'  +  A, 

F  =  the  crushing  strength  of  the  material,  or  stress  at  the  yield 
point.  This  is  the  maximum  intensity  of  stress  in  the 
column  when  the  mean  intensity  of  stress  is^. 

f  =  the  intensity  of  working  stress  in  the  column  (  =  F-r-  ri). 
This  is  the  maximum  intensity  of  stress  in  the  column 
when  the  mean  intensity  of  stress  is^'. 

m  =  a  coefficient  for  the  end  conditions. 

For  end  conditions  as  in  Table  VIII  (Unwin,  page  80)  : — 

I.  Fixed  at  one  end  a.\\d.free  at  the  other,         -         -     m  =  ^  ; 
II.   "  Pin  ended  "  (both  ends  free  but  guided),       -         m  =  i  ; 

III     "  Pin  and  square"  (one  end  fixed  the  other  guided).  m  =  \\ 

IV.    "  Square  ended  "  (both  ends  fixed),    -         -         -     m  =  4. 

The  diagram  of  Fig.  7  is  for  the  ultimate  resistance  of  pin  ended 
columns  with  a  material  having  a  crushing  resistance,  F,  (.yield 
point)  of  36,000  pounds  per  square  inch,  and  a  modulus  of  elas- 
ticity, E.  of  29,400,000.  The  value  of  p  is  36,000  for  a  very  short 
compression  member,  and  it  is  evident  that  a  long  column  could 
not  be  expected  to  have  a  greater  strength  ;  hence  no  formula 
should  be  used  which  would  give  a  value  of  p  in  excess  of  the 
crushing  resistance  F.  Referring  to  the  diagram,  it  will  appear 
that  the  Euler  formula  (represented  by  the  curve  E  E\  E^)  cannot 
apply  to  columns  (of  this  particular  material)  in  which  /  H-  p  <  90. 
If  columns  with  a  ratio  of  /  to  p  less  than  this  limit  yielded  by 
simple  crushing,  and  those  with  a  greater  ratio  of  /  to  p  followed 
Euler's  formula,  the  straight  line  FFl  and  the  curve  F^ElE.t 
would  give  the  laws  for  all  lengths  of  columns.  It  is  not  reason- 
able to  expect  such  an  abrupt  change  of  law  in  passing  this  limit 
(/  -r-  p  =-  90)  ;  and,  as  already  stated,  columns  of  moderate  length 
fail  under  a  mean  stress  considerably  less  than  the  simple  crush- 
ing resistance  of  the  material;  or  the  strength  of  columns  is  in- 
versely as  some  function  of  the  length  divided  by  the  diameter. 

Mr.  Thomas  H.  Johnson   has  developed  a  formula  which  is 


based  on  the  assumption  that  the  strength  of  the  column  may  be 
taken  inversely  as  /  -f-  p.     This  expression  is 


in  which  the  coefficient  k  has  the  value, 

*=^  H4Z: 

3\3  miS  E. 

This  formula  is  represented  by  the  straight  line  THJ^  in  Fig. 
7.  It  will  be  noted  that  this  line  is  tangent  to  the  Euler  curve  at 
J.2,  and  the  equation  of  the  latter  is  to  be  used,  should  the 
columns  exceed  the  length  corresponding  to  this  point  of  tan- 
gency,  (/-j-p  >  150).  This  expression  is  very  simple,  after  k 
has  been  determined.  It  is  very  convenient  in  making  a  large 
number  of  computations  for  columns  of  any  one  material,  and  it 
is  employed  in  bridge  design  to  a  considerable  extent.  It  does 
not  appear  to  have  any  advantage,  on  the  ground  of  simplicity, 
when  some  particular  value  of  k  does  not  apply  to  several  compu- 
tations. Furthermore,  this  formula  gives  rather  large  sections  for 
columns  in  which  I  -i-  p  is  less  than  about  40. 

For  determination  of  nominal  working  stress,  p  (as  computed 
above)  may  be  divided  by  a  suitable  factor  of  safety,  «.  Or  if 
p-±n=p',  the  expression  may  be  put  in  the  following  form  for 
direct  computation  of  mean  working  stress  : 


TT'  E  p  . 

Professor  J.  B.  Johnson  has  derived  a  formula  from  the  results 
of  the  very  careful  experiments  of  Considere  and  Tetmajer.  His 
formula  is  : 

(3) 

for  pin  ended  columns.  The  curve  J B Jl  (Fig.  7)  represents 
this  expression.  This  curve  is  a  parabola  tangent  to  the  Euler 
curve,  and  with  its  vertex  in  the  axis  or  ordinates  at  F,  the 


-29  — 

direct  crushing  stress  of  the  material.  For  columns  having 
/-7-p  greater  than  the  value  corresponding  to  the  point  of  tan- 
gency  J:,  (should  such  be  used)),  the  Euler  formula  is  to  be  em- 
ployed. This  formula  of  Professor  Johnson's  is  empirical,  but  it 
agrees  remarkably  well  with  very  refined  experiments  on  break- 
ing loads.  It  gives  considerably  higher  values  for  allowable 
stress  than  other  generally  accepted  formulas,  probably  because 
it  is  based  upon  more  refined  tests,  or  upon  conditions  further 
removed  from  those  of  practice. 

Professor  Johnson  says  (Materials  of  Construction,  pages 
301-302)  that  both  Bauschinger  and  Tetmajer  "mounted  their 
columns  with  cone  or  knife-edge  bearings  at  the  computed  gravity 
axis,  while  M.  .Considere  mounted  his  with  lateral-screw  adjust- 
ments, and  arranged  a  very  delicate  electric  contact  at  the  side 
so  as  to  indicate  a  lateral  deflection  as  small  as  o.ooi  mm.  He 
then  applied  moderate  loads  to  the  columns  and  adjusted  the  end 
bearings  until  they  stood  under  such  loads  rigidly  vertical,  with 
no  lateral  movement  whatever."* 

It  would  appear  that  this  precaution  tends  to  make  the  test  one 
of  the  material  and  not  of  a  long  strut ;  for  the  eccentricity  of  the 
load  (relative  to  the  nominal  geometric  axis)  compensates,  in  a 
measure,  for  the  lack  of  homogeniety  of  the  material.  Had  the 
correction  been  made  under  greater  load,  the  results  of  the  tests, 
if  plotted  in  Fig.  7,  would  probable  be  still  nearer  the  line  F F^ 
and  the  difference  between  these  test  columns  and  columns  as 
used  in  practice  would  be  greater,  requiring  a  higher  contingency 
factor  in  the  latter  for  safety. 

For  determining  the  working  stress,  the  value  of  p  (as  com- 
puted from  the  above  form  of  Johnson's  expression)  should  be 
divided  by  a  suitable  factor  of  safety  n.  Or.  the  formula  may  be 
put  in  the  following  form  for  computing  nominal  working  stress  : 

(4) 


*"This  precaution  is  essential  to  a  perfect  test  of  the  material    * 
Only  in  this  way  can  other  sources  of  weakness  be  eliminated." — [J.  B.  J.] 


—  30  — 

The  Rankine,  or  Gordon,  formula  (see  Church's  Mechanics, 
pages  372-376)  has  been  extensively  used  for  columns.  It  may 
be  expressed  as  follows  : 

P_=  F 

A  \+^uy-          (5) 


The  above  formula  is  based  upon  experiments  on  the  breaking 
strength  of  columns.  The  coefficient  (3  is  purely  empirical,  and 
this  fact  limits  its  usefulness,  for  it  leaves  much  uncertainty  as  to 
how  this  coeffcient  should  be  modified  for  different  materials  than 
those  which  have  been  actually  tested  as  columns.  The  meaij 
intensity  of  working  stress,  />',  might  be  inferred  by  dividing /by 
n,  or  the  expression  can  be  written  ; 

/ 

(6) 


but  it  is  not  entirely  satisfactory  to  assume  the  action  for  stresses 
within  the  elastic  limit,  from  the  results  of  tests  for  breaking 
strength.  The  form  of  the  Rankine  expression  is  rational,  but 
the  coefficient  /?  is  not. 

Professor  Merriman  says,  in  his  Mechanics  of  Materials,  page 
129:  "Several  attempts  have  been  made  to  establish  a  formula 
for  columns  which  shall  be  theoretically  correct  .  .  .  The  most 
successful  attempt  is  that  of  Ritter,  who,  in  1873,  proposed  the 
formula 


(7) 


"The  form  of  this  formula  is  the  same  as  that  of  Rankine's 
formula,  .  .  .  but  it  deserves  a  special  name  because  it  completes 
the  deduction  of  the  latter  formula  by  finding  for  /3  a  value 
which  is  closely  correct  when  the  stress  /  does  not  exceed  the 
elastic  limit  F."  The  above  notation  is  changed  to  agree  with 
that  previously  used  in  this  article.  The  ratio  /r-r-/is  the  factor 
of  safety.  For  ultimate  strength,  this  formula  might  be  written  : 


-31- 


but  the  first  form  (eq.   7)  is  the   more  important.      The    curve 
Rl  T  RI    (Fig.    7)    is   the    graphical    representation    of  the    last 


expression,  eq.  8.* 

Merriman  gives  the  Euler  formula  for  a  factor  of  safety  of  n  —  - 
F-s-f,  which  is 

,=,£_£«„(£)•  (9) 

Failure  occurs  if  />  F.  The  Ritter  formula  (eq.  8)  reduces  to 
this  last  expression  for  columns  so  long  that  the  term  unity  in  the 
denominator  is  negligible  ;  strictly  speaking,  this  is  only  the  case 
when  /  -r-  p  =  infinity.  Professor  Merriman  also  shows,  mathe- 
matically, that  the  two  curves  E  E^  E.2  and  Rl  T  R2  are  tangent  to 
each  other  when  /  H-  p  —  infinity. 

If  /  -r-  p  =  o,  the  Kilter  formula  reduces  to  p'  =  P'  -f-  A  ,  which 
is  the  ordinary  formula  for  short  compression  members. 

The  fact  that  this  formula  is  rational  in  form,  that  it  gives  the 
correct  values  at  the  limits  /-*-  p  =  oo  and  /H-  p  =  o,  and  that  it 
lies  wholly  within  the  boundary  FFlElE.i  (Fig.  7)  all  justify  its 
use.  and  it  will  be  adopted  in  this  work.  It  will  be  noted  from 
Fig.  7  that  the  Ritter  and  Rankine  formulas  agree  very  closely 
for  the  material  taken  for  illustration  ;  but  the  fact  that  the  curve 
of  the  latter  crosses  the  Euler  curve  near  the  right  hand  limit  of 
the  diagram  indicates  that  its  constant  fi  is  not  theoretically 
correct. 

All  of  the  above  formulas  give  the  value  of  the  mean  ultimate 
stress  (p  =  P-r-  A),  or  the  mean  working  stress  (p'  =  P'  -*-  A), 
corresponding  to  a  maximum  ultimate  stress  F,  or  a  maximum 
working  stress  f,  respectively.  However,  the  ordinary  problem 
of  design  is  to  assign  proper  dimensions  for  the  member  under 

^Professor  Merriman  developed  equation  8,  independently,  but  later  than 
Ritter.  He  gives  Ritter  sole  credit  for  the  formula  in  the  recent  (1897)  edi- 
tion of  his  Mechanics  of  Materials. 


the  giver,  load.  It  is  not  practicable  to  solve  directly  for  the  area  in 
such  expressions  as  those  given  in  this  article  as  p'  (or  p)  and  /> 
are  both  functions  of  the  area  of  the  cross  section.  It  is  usual  to 
assume  a  section  somewhat  larger  than  that  demanded  for  simple 
crushing,  and  then  to  check  for  the  ultimate  load  />,  or  the  work- 
ing load  P'.  Mr.  W.  N.  Barnard  has  devised  a  diagram  which 
is  very  convenient  for  these  computations.  It  is  shown,  to  a  re- 
duced scale,  in  Fig.  8.  The  four  curves  are  for  the  four  end  con- 
ditions given  on  page  27  (or  Unwin,  Table  VIII,  page  80).  They 
are  plotted  for  a  maximum  working  stress  of  10,000  pounds  per 
square  inch  ;  but  may  be  used  for  any  other  stress  by  proceeding 
as  follows  :  Assume  a  trial  cross-section,  which  fixes  p.  Divide 
/  by  this  value  of  p  ;  take  this  quotient  on  the  lower  scale  and  pass 
directly  upward  to  the  proper  curve  for  the  given  end  conditions  ; 
then  pass  horizontally  to  that  one  of  the  radiating  diagonals 
which  is  numbered  to  correspond  with  the  selected  stress  ;  from 
this  last  point  pass  upward  to  the  horizontal  scale  at  the  top  of 
the  diagram,  where  the  value  of  the  unit  load  or  mean  working 
stress,  (/'),  is  read  off-1  If  this  value  of/'  agrees  sufficiently  well 
with  the  quotient  of  the  load  divided  by  the  trial  area,  the  section 
may  be  considered  as  satisfactory. 

In  the  case  of  a  square-ended  column,  or  when  the  supporting 
action  of  the  ends  is  equal  in  all  possible  planes  of  flexure,  it  is 
sufficient  to  take  the  least  radius  of  gyration  of  the  section  ;  or  to 
take  p  for  the  axis  about  which  the  section  is  weakest.  In  case 
of  a  pin-ended  column,  as  a  connecting  rod,  the  cylindrical  sup- 
porting pins  make  it  equivalent  to  a  square-ended  column  against 
flexure  in  the  plane  of  the  axes  of  the  pins,  provided  these  bear 
symmetrically  with  reference  to  the  axis  of  the  column  ;  while 
the  column  is  pin-ended  with  reference  to  a  plane  perpendicular 
to  the  axes  of  the  pins.  If  the  cross-section  of  such  a  column 
has  equal  dimensions  in  these  two  planes  (circular,  square  sec- 
tions, etc.),  the  column  need  only  be  computed  for  the  latter 


1  The  method  of  using  the  diagram  is  indicated  by  the  arrows,  for  an  ex- 
ample in  which  /  -f-  p  =  80  and  the  maximum  working  stress  =  14,000.  In 
this  case,  p'  is  found  to  be  about  7,900. 


-33- 

piane.  If  the  pin-ended  column  has  an  oblong  section  (elliptical, 
rectangular  but  not  square,  I  section,  etc.),  it  may  be  weaker  in 
either  of  these  two  planes,  notwithstanding  the  difference  in  end 
conditions  relative  to  them  ;  and  it  may  be  necessary  to  compute 
for  both  planes,  unless  the  section  is  obviously  stronger  in  one  of 
them.  If  a  rectangular,  or  elliptical,  column  has  a  section  in 
which  the  dimension  in  the  plane  of  the  pins  is  more  than  one 
half  the  dimension  in  the  plane  perpendicular  to  the  pins,  it  will 
suffice  to  compute  as  a  pin-ended  column  against  flexure  in  the 
latter  plane,  and  vice  versa.  In  the  preceding  discussion,  the 
various  formulas  have  been  given  both  for  breaking  and  for  work- 
ing loads.  The  Euler  and  Ritter  formulas  are  derived  from  the 
theory  of  elasticity  ;  hence  these  are  proper  for  computations  per- 
taining to  working  loads,  in  which  the  stress  should  never  exceed 
the  elastic  limit  *  It  does  not  follow  that  these  two  rational  for- 
mulas will  agree  with  experiments  on  the  ultimate  resistance  of 
columns.  These  expressions  are,  in  this  respect,  like  the  com- 
mon beam  formulas.  Such  formulas  as  Rankine's  and  J.  B. 
Johnson's,  derived  from  tests  of  ultimate  resistance  of  columns, 
are,  for  similar  reasons,  less  rigidly  applicable  to  working  loads 
and  stresses. 

15.  Eccentric  Load.  Tension,  or  Compression,  Com- 
bined with  Bending.— [Unwin,  §  43,  pages  89-90.] 

It  is  not  practicable  to  solve  the  equation 

/=£  +  - 

for  the  direct  determination  of  the  dimensions  of  cross  section  to 
sustain  a  given  eccentric  load  (/>)  with  an  assigned  intensity  of 
stress  (f),  because  both  A  and  Zare  functions  of  the  required 
dimensions.  With  solid  square  or  circular  sections,  or  in  general 
when  only  one  dimension  is  unknown,  it  is  possible  to  reduce  the 
above  equation  to  a  form  which  can  be  solved  for  this  unknown 
quantity  ;  but  the  algebraic  expression  is  a  troublesome  cubic 


*The  Euler  formula  is  not  applicable  for  practical  applications,  except  for 
quite  long  columns. 


—  34  — 

equation.  The  practical  method  is  to  assume  a  trial  section  and 
check  this  for  either  Povf. 

Example  i.  A  small  crane  (Fig.  ga)  has  a  clear  swing  of  28 
inches.  The  section  at  m  .  .  n  is  shown  by  Fig.  90.  Find  the 
load  corresponding  to  a  maximum  fibre  stress  (compression)  of 
9000  pounds  per  square  inch  at  n. 

sP       Pr    .     p==    fAZ 

A  ^   Z    '  Z+rA' 

r=-  28  -i-  2  =  30;     ^4  — 2X4—  1.5X3  =  3.5; 

(  2  X  64  -  1.5  X  27  )  =  3.65  (See  Unwin,  p.  58). 

.    p=  9000  X^. 5  X  3,65  =  Io6o  lbs< 
3.65  +  30  X  3-5 

Example  2.  A  punching  machine  (Fig.  loa)  has  a  reach  of 
22".  Maximum  force  (P)  acting  at  the  punch  is  taken  at  70,000 
Ibs.  Design  section  m  .  .  n  so  that  maximum  fibre  stress  at  n  (ten- 
sion) shall  be  about  2,400  pounds  per  square  inch. 

The  general  form  of  section  best  adapted  for  this  case  is  that 
shown  in  Fig.  lob  Taking  the  trial  dimensions  as  in  Fig.  iob, 
the  neutral  axis  is  found  to  be  8"  from  n. 

.'.  r=  22  +  8  =  30.  It  is  also  found  that  A  =  216,  and  Z  = 
960 

,  /i=  70000  +  70090^0  =  32J 

This  is  slightly  greater  than  the  limit  assigned  for  maximum  in- 
tensity of  working  stress.  If  this  excess  is  not  considered  per- 
missible, a  somewhat  stronger  section  is  to  be  taken,  checking 
the  latter  if  this  step  seems  necessary. 

16.  Combined  Torsion  and  Flexure. — [Unwin,  $  44,  pages 
90-91.] 

The  expression  (Unwin,  eq.  27,  page  90), 


T  (i) 

may  be  obtained  from  the  relation 


—  35- 

/.-*[/+%'/*'+  4/fl*  (*) 

in  which  fu  =  the  maximum  intensity  of  normal  stress  due  to 
the  combined  bending  and  twisting  moments  ;  f=  the  intensity 
of  stress  due  to  the  simple  bending  moment  ;  a.ndfs  =  the  inten- 
sity of  shearing  stress  due  to  the  simple  twisting  moment. 


[7p  =  2  /,  for  circles,  or  other  sections  in  which  the  moments  of 
inertia  about  two  perpendicular  axes  are  each  equal].     Hence, 


VM*+T       (3) 

Since  M  =•*—  -  and  T=  2-^3    ,    for   an    equal  intensity  of  stress 

(f~f*)  in  bending  or  twisting,  T  '=  2  M,  for  the  same  section. 
It  is,  therefore,  allowable  to  substitute  for  MK  (the  bending  mo- 
ment equivalent  to  the  combined  bending  and  twisting  moments) 
\  TK  (the  twisting  moment  equivalent  to  the  combined  bending 
and  twisting  moments).  Then 

</  M'1  +  T\ 


.  •  .     re  =  M  +  v/  M  2  +  T\  (4) 

In  a  similar  way,  eq.  (2ya)  of  Unwin,  may  be  reduced  to 

Te  =.  2  MK  =  |  M+  f  v/  M1  +  Tl  (5) 

The  expressions  designated  above  as  equations  (3)  and  (4),  are 
two  forms  of  Rankine's  formula  for  combined  bending  and  twist- 
ing, while  equation  (273)  [Unwin]  and  the  above  derived  form 
eq.  (5),  are  due  to  Grashof.  Some  authorities  use  the  latter,  but 
the  Rankine  formula  is  adopted  by  Unwin,  and  by  many  others, 
and  it  will  be  followed  in  the  present  work. 

*  See  Church's  Mechanics,  eq.  (8),  page  317,  in  which  q  max  =./„,  p=f, 
and/>s=y~s,  as  in  eq.  (2)  above. 


—  36- 

If  the  ratio  of  M  to  T  be  called  k,  M  =  k  7';  hence  eq.  (4)  re- 
duces to 

7;  =  |>4-X/T8  +  i]  T  (6) 

and  eq.  (5)  reduces  to 

7;  =  [f£+fx//r+i]r  (7) 

Convenient  graphical  solutions  of  equations  (6)  and  (7)  are 
shown  in  Fig.  n  and  12,  respectively. 

For  solution  of  the  Rankine  formula,  eq.  (6),  make  O  a  =  uni- 
ty (Fig.  n);  lay  off  O  b  =  k,  to  the  same  scale  on  the  vertical 
axis  O  V  ;  draw  a  b,  extending  it  beyond  b  for  a  length  somewhat 
greater  than  k  ;  then,  a  b  =  </  ~ol?  +  O~a*  =  ^  k1  +  i.  Next, 
lay  off  b  c  =  O  b  =  k  (along  the  extension  of  a  b}  ;  now,  b  c  +  a  b  = 
a  c  =  k  +  V  "&~+~i  ;  hence,  Te  =  (a  c)  T. 

For  solution  of  the  Grashof  formula,  eq.  (7),  make  O  a  —  unity, 
O,  a  =  |  ,  and  O,a  =  J  ;  lay  off  Ob-^k,  hence  a  b  --=  V  &  +  i, 
and  a  b.L  =  £  \/  k1  +  i.  Now,  since  Ol  b^  =  f  k,  if  b.2  c  be  laid  off 
equal  to  O}  bl  ; 

ac=b.ic+  ab.,=lk  +  ^  ^"F'+T  ;  hence  Tc  =  (a  c)  T. 

Example  i.  An  engine  is  16"  X  24"  (piston  16  inches  in  diam- 
eter and  stroke  of  24  inches),  steam  pressure  =  100  pounds  per 
sq.  inch.  The  centre  of  the  crank  pin  overhangs  the  centre  of 
the  main  journal  by  15"  (measured  parallel  to  axis  of  the  shaft). 
Assume  that  the  pressure  on  the  crank  pin  may  be  equal  to  100 
pounds  unbalanced  pressure  per  sq.  inch  of  the  piston  when  the 
connecting  rod  is  perpendicular  to  the  crank  radius.  Allowing 
8000  pounds  as  the  maximum  fibre  stress  in  the  shaft  ;  compute 
its  diameter. 

Area  of  piston  =  200  sq.  inches  ;  radius  of  crank  (arm  of  max- 
imum twisting  moment)  =  12";  arm  of  bending  moment  =  15". 

.'.  T=  200  X  100  X  12  =  240,000  inch  pounds.  Also,  k  =  15 
-T-  12  =  1.25. 


10 


I  1.25  -f  1.601240,000  =  684,000  inch  pounds. 


37 


17.   Combined  Torsion  and  Compression.  — 
Propeller  shafts  of  steamers  and  vertical  shafts  carrying  consid- 
able  weight  are  subjected  to  combined  twist  and  thrust.     The 
span,  or  distance  between  bearings,  is  frequently  so  small  that  the 
shaft  may  be  considered  as  subjected  to  simple  compression,  so 
far  as  the  action  of  the  thrust  is  concerned. 
The  intensity  of  this  compressive  stress  is 


in  which   IV  =  the  thrust,  and  d  =  the  diameter  of  the  (solid  cir- 
cular) shaft. 

If  T=  the  twisting  moment  on  the  shaft,  r  =•  \  d  --  the  radius 
of  the  shaft,  7p  =  the  polar  moment  of  inertia  =  2  /(the  rectan- 
gular moment  of  inertia),  and_/g  —  the  intensity  of  shearing  stress 
due  to  T,  then 

r==X/p  =  4/7    .    f=Td=  i6_T 

r  d  47        TT^' 

for  solid  circular  shafts. 

The  resultant  maximum  stress  is  that  due  to  the  combined  ac- 
tions of  a  normal  stress  (compression)  and  a  tangential  stress 
(shear),  as  in  the  case  of  combined  bending  and  torsion  (art.  16)  ; 
hence,  equation  (2)  of  the  preceding  article  may  be  used  to  find 
the  equivalent,  intensity  of  stress,  or 


209403 


-38- 

If  the  value  of  /n  is  assigned,  and  d  is  to  be  found  for  given 
values  of  J^and  7]  it  is  possible  to  solve  the  transformed  (cubic) 
equation  ;  but  it  is  much  more  convenient  to  assume  a  trial  value 
of  d  (somewhat  larger  than  that  required  for  the  twisting  moment 
alone)  and  then  to  check  for/,,. 

If  the  span,  between  bearings,  is  so  great  that  the  shaft  must 
be  considered  as  a  column  liable  to  buckle,  the  value  of  the  mean 
intensity  of  compressive  stress,  (/>'),  may  be  found  by  eq.  (7)  of 
art.  14. 

Since  p'  is  the  mean  intensity  of  compressive  stress  in  the  long 
column  which  corresponds  to  a  maximum  intensity  of  stress/, 
a  short  compression  member  of  the  same  cross- section,  would 
be  capable  of  standing  a  load  IV}  greater  than  W  in  the 

ratio  of/ to/  ;  or  W,  =  *  W.     This  value  of  Wl  is  tojbe  substi- 

P 
tuted  for  W\\\  eq.  (i)  of  this  article,  for  shafts  of  long^span. 


III. 

SPRINGS. 


18.  Distinguishing  Characteristic  of  Springs. — Springs  are 
characterized  by  a  considerable  distortion  under  a  moderate  load. 
Every  machine  member  is,  in  a  sense,  a  spring,  for  no  material  is 
absolutely  rigid  and  the  application   of  a  load   always  produces 
stress   and  accompanying  strain.     By  proper  selection   and  dis- 
tribution of  material  it  is  possible  to  control  (within  wide  limits) 
the  degree  of  distortion  under  a  given  load. 

An  absolutely  rigid  material  would  be  practically  unfit  for  the 
construction  of  any  member  subject  to  other  than  a  perfectly 
quiescent  load  ;  for  (as  shown  in  art.  8)  the  stress  due  to  a  sud- 
denly applied  load  would  be  infinite  if  the  corresponding  distor- 
tion of  the  member  were  zero. 

While  it  is  usually  desirable  to  restrict  the  distortions  of  most 
machine  parts  to  very  small  magnitudes,  there  are  many  cases  in 
which  considerable  distortion  under  moderate  load  is  desirable  or 
essential.  To  meet  this  last  requirement  the  member  is  often 
given  some  one  of  the  forms  commonly  called  springs. 

19.  The  Principal  Applications  of  Springs.— Springs  are  in 
common  use  : 

I.  For  weighing  forces  ;  as  in  spring  balances,  dynamometers, 
etc. 

II.  For  controlling  the  motions  of  members  of  a  mechanism 
which  would  otherwise  be  incompletely  constrained  ;  for  example, 
in  maintaining  contact  between   a  cam   and   its  follower.     This 
constitutes  what  Reuleaux  has  called  ''force  closure". 

III.  For  absorbing  energy  due  to  the  sudden  application  of  a 
force  (shock)  ;  as  in  the  springs  of  railway  cars,  etc. 

IV.  As  a  means  of  storing  energy,  or  as  a  secondary  source  of 
energy  ;  as  in  clocks,  etc. 


An  important  class  of  mechanisms  in  which  springs  are  used  to 
weigh  forces  is  a  common  type  of  governor  for  regulating  the 
speed  of  engines  or  other  motors.  In  those  governors  which  use 
springs  to  oppose  the  centrifugal,  or  other  inertia  actions,  the 
springs  automatically  weigh  forces  which  are  functions  of  speed, 
or  of  change  of  speed.  The  links,  or  other  connections,  which 
move  relative  to  the  shaft  with  any  variation  of  the  above  forces, 
correspond  to  the  indicating  mechanism  of  ordinary  weighing  de- 
vices. 

The  first  of  the  above  mentioned  applications — the  weighing  of 
forces — is  usually  the  most  exacting  as  to  the  relation  between 
the  load  and  the  distortion  of  the  spring  throughout  the  range  of 
action.  In  the  second  and  third  classes  of  application,  it  is  fre- 
quently only  required  that  the  maximum  load  and  distortion  shall 
lie  within  certain  limits,  which  often  need  not  be  very  precisely 
defined.  The  use  of  springs  for  storing  energy  (as  the  term 
spring  is  ordinarily  understood)  is  almost  wholly  confined  to 
light  mechanisms  or  pieces  of  apparatus  requiring  but  little  power 
to  operate  them. 

20.  Materials  of  Springs. — Springs  are  usually  of  metal  ; 
although  other  solid  substances,  as  wood,  are  sometimes  used. 
A  high  grade  of  steel,  designated  as  spring  steel,  is  the  most 
common  material  for  heavy  springs,  but  brass  (or  some  other 
alloy)  is  often  used  for  lighter  ones. 

A  confined  quantity  of  air,  or  other  compressible  fluid,  is  used 
in  many  important  applications  to  perform  the  office  of  a  spring. 
The  air-chamber  of  a  pump  with  its  inclosed  air  is  a  familiar  ex- 
ample of  what  may  be  called  a  fluid  spring  used  to  reduce  shock 
("  water  hammer  ").  The  characteristic  distortion  of  the  solid 
springs  is  a  change  in  form  rather  than  of  volume  ;  while  the 
fluid  springs  are  characterized  by  a  change  of  volume  with  inci- 
dental change  of  form 

Soft  rubber  cushions,  or  buffers,  are  not  infrequently  employed 
as  springs,  and  these  are  in  some  respects  intermediate  in  their 
action  to  the  two  classes  mentioned  above.  It  is  usually  not 
necessary,  in  these  simple  buffers,  or  cushions,  to  secure  a  very 


Fig.  2  7. 


exact  relation  between  the  loads  and  the  distortions  under  such 
loads.  The  discussion  of  the  confined  gases  (fluid  springs)  is  not 
within  the  scope  of  the  present  work ;  hence  the  following  treat- 
ment will  be  limited  to  solid  springs. 

21.  Forms  of  Solid  Springs  — Springs  may  be  subjected  to 
actions  which  extend,  shorten,  twist,  or  bend  them,  producing 
stresses  in  the  material,  the  character  of  which  depend  upon  both 
the  form  of  the  spring  and  upon  the  manner  of  applying  the  load. 

I.  Flat   Springs  are  essentially  beams,   either  cantilevers,  or 
with  more  than    one  support.     These   springs   are   subjected    to 
flexure  when  the  load  is  applied,  and  the  resultant  stresses  are 
tension  in  certain   portions  of  the  material,  and  compression   in 
others,  with  a  transverse  shear,  as  in  all   beams  ;  the  shear  may 
usually  be  neglected  in  computations.     The  ordinary  beam  for- 
mulas for  strength  and  rigidity   may  be  applied  to  flat  springs, 
with  constants  appropriate  to  the  particular  material  and  form  of 
beam  used. 

Flat  springs  may  be  simple  prismatic  strips,  of  uniform  cross- 
section,  (Figs.  13  or  16)  ;  although  it  is  preferable  that  the  form  of 
such  springs  approximate  those  of  the  ' '  uniform  strength  ' '  beams 
(Figs.  14  or  15  ;  17  or  18). 

It  is  often  desirable  or  practically  necessary  to  build  up  these 
springs  of  several  layers,  leaves,  or  plates,  producing  a  laminated 
spring.  It  will  appear  from  the  discussion  of  these  laminated 
springs  that  they  may  be  properly  treated  as  a  modification  of  one 
form  of  "uniform  strength  "  beam.  The  neutral  surface  of  the 
beam  used  as  a  spring  may  be  initially  curved,  either  to  clear 
other  bodies,  or  to  give  the  spring  an  advantageous  form  when  it 
is  under  normal  load.  See  Fig.  21. 

Two  or  more  springs  may  be  compounded,  as  in  the  "  ellipti- 
cal "  springs  or  in  the  platform  springs  frequently  used  under 
carriages.  In  such  cases,  each  spring  may  be  computed  sepa- 
rately, and  the  total  deflection  is  the  sum  of  the  deflections  of  the 
separate  springs  of  the  set. 

II.  Helical,  or  Coil  Springs  are  most  commonly  used  to  resist 
actions  which   extend,  shorten,  or  twist  the  spring  relatively  to 


42 

its  longitudinal  axis.  These  are  sometimes  improperly  called 
spiral  springs. 

The  stress  in  the  wire  (or  rod)  of  which  a  helical  spring  is 
made  is  somewhat  complex,  consisting  of  torsion  combined  with 
tension  or  compression,  or  both.  In  a  "  pull  spring  ",  one  which 
is  extended  longitudinally  under  the  load,  the  predominating 
stress  (with  ordinary  proportions)  is  a  torsion,  and  there  is  a 
secondary  tensile  stress  in  the  wire.  In  a  "push  spring",  one 
which  is  shortened  by  the  load,  the  predominating  stress  is  tor- 
sion, with  a  secondary  compressive  stress.  When  the  helical 
spring  is  subjected  td  an  action  which  twists  the  spring  (as  a 
whole)  the  principal  stress  in  the  wire  is  that  due  to  flexure 
(tension  and  compression  in  opposite  fibres)  and  the  secondary 
stress  is  torsion. 

Helical  springs  are  sometimes  arranged  in  "  nests",  springs  of 
smaller  diameter  being  placed  within  those  of  larger  diameter. 
In  these  cases,  the  different  springs  of  a  set  are  computed  sepa- 
rately. This  last  arrangement  is  common  practice  in  car  trucks. 

III.  Spiral  Springs,  properly  so  called  are  those  of  the  form  of 
the  familiar  clock  spring.     These  are  best  adapted   for  a  twist 
relative  to  the  axis  of  the  spiral,  and  are  usually  employed  when 
a   very  large  angle  of  torsion   between    the  two  connections   is 
necessary.     In  this  form  of  spring,  the  stress  in   the  material  is 
that  due  to  flexure  ;  or  tensile  and  compressive  stress  on  opposite 
sides  of  the  neutral  axis. 

IV.  ffelico- Spiral  Springs.     The  form  of  spring  represented  by 
the  common  upholstery  spring  may  be  looked  upon  as  a  spiral 
spring  which  has  been  elongated,  and  given  a  permanent  set,  in 
the  direction  of  its  axis  ;  or  it  maj'  be  considered  as  a  modified 
helical  spring  in  which  the  radii  of  the  successive  coils  are  not 
equal.     It  is  thus  intermediate  between  the  two  preceding  classes. 
This   last   form   is   not  usual   in  machine  construction  ;  though 
it  has  the  advantage  over  the  common  helical  spring  of  consider- 
able lateral  resistance,   and   it  may  be  employed  to  advantage 
where  it  is  difficult  or  undesirable  to  otherwise  constrain   the 
spring  against  buckling.     This  spring  is  only  used  as  a  push 


43 

spring,  to  resist  a  compressive  action.  The  springs  used  on  the 
ordinary  disc  valves  of  pumps  are  often  of  this  form,  as  they 
will  close  up  flat  between  the  valve  and  guard.  Car  springs  are 
sometimes  made  of  a  flat  strip  or  ribbon  of  steel  wound  in  this 
general  form,  with  the  edges  of  the  strip  parallel  to  the  axis  of  the 
spring. 

V.  Occasionally  straight  rods,  usually  of  circular  or  rectangular 
cross-sections,  are  employed  to  resist  torsion  relative  to  their 
longitudinal  axis.  These  are  comparatively  stiff  springs,  and  the 
stress  is,  of  course,  torsional.  Every  line  of  shafting  is  necessarily 
a  spring,  in  this  sense. 

The  following  summary  gives  the  ordinary  forms  of  solid 
springs;  the  kinds  of  loading  to  which  they  are  subjected;  and 
the  predominating  stresses  resulting  from  the  different  loads. 

GENERAL  SUMMARY  OF  SPRINGS. 


FORM  OF  SPRING.  LOAD  ACTION.  PREDOMINATING  STRESS. 

Flat  Spring.  Flexure  or  Bending.  Tension  and  Compression. 

Helical  Spring.  Extension,  Pull.  iTorsion  (plus). 

Compression,  Push.  Torsion  (minus). 

Torsion,  Twist.  Tension  and  Compression. 

Spiral  "  Torsion,  Twist.  Tension  and  Compression. 

22.  Computations  of  Simple  Flat  Springs. — The  following 
notation  will  be  used  in  treating  of  flat  springs  with  rectangular 
cross-sections  : 

y°==load  applied  to  the  spring. 

/,  —  free  length  of  the  spring. 
f—  intensity  of  stress  in  outer  fibres. 

/—  moment  of  inertia  of  most  strained  section. 

h  =  dimension  of  this  section  in  plane  of  flexure. 

b  =  dimension  of  this  section  normal  to  plane  of  flexure. 

E  -—  modulus  of  elasticity  of  material. 

8  =  deflection  of  the  spring. 

The  six  forms  of  rectangular  section  beams,  shown  by  Figs.  1310 
18,  are  the  most  important  of  those  used  as  simple  flat  springs. 


44 

These  will  be  designated  Type  I,  II,  etc.,  as  in  the  following  table, 
which  gives  the  constants  to  be  substituted  in  the  general  for- 
mulas for  computations  relating  to  each  type. 

TABLE. 


TYPE 


COEFFICIENTS. 


B 


i 

* 

* 

f 

i 

f 

1 

f 

2 

A 

The  theory  of  strength  against  flexure  gives  :  For  rectangular 
section  beams  supported  at  the  ends  and  loaded  at  the  middle. 
(Types  I,  II,  III). 

(i) 


. 

46 

For  rectangular  section  cantilevers,  with  load  at  free  end, 

PL^1  fbh*  (2) 

6 

Or  the  general  formula  for  the  strength  of  rectangular  section 
beams  may  be  written 

PL=Afbhl  (3) 

In  which  the  coefficient  A  has  the  values  given  in  the  Table. 
The  theory  of  elasticity  of  beams  gives 


45 


or  for  rectangular  cross  sections 


In  which  /?  and  ^  are  as  given  in  the  Table,  for  the  types  under 
consideration. 

The  last  equation  (5)  may  be  used  for  all  computations  as  to 
rigidity  of  fiat  springs  (beams),1  provided  the  elastic  limit  is  not 
exceeded.  The  only  constant  for  the  material  which  enters  this 
expression  is  the  modulus  of  elasticity  (E}  ;  this  is  simply  the 
ratio  of  stress  to  strain  which  holds  up  to,  but  not  beyond,  the 
elastic  limit  ;  hence  any  computation  made  by  this  formula 
should  be  checked  for  safety.  Equation  (3)  may  be  used  for  this 
purpose.  To  illustrate,  assume  that  a  rectangular  section  pris- 
matic spring  (Type  I)  has  a  length  between  supports  of  L  —  30"  ; 
the  load  at  the  middle  is  P=  1000  Ibs.  ;  the  deflection  under  this 
load  is  to  be  8=  1.5  inches  ;  and  the  spring  is  made  of  a  single 
strip  of  steel  f  inch  thick  (h).  Required  the  breadth  (b}  of  the 
spring,  assuming  the  modulus  of  elasticity,  .£=  30,000,000. 

From  eq.  (5)  :  — 

b  =  B  PL*  =  I  x  i_^ooX_27,ooo_X5i2  =  2  g        inches 
Edh        4     30,000,000  X  1.5  X  27 

This  gives  the  width  of  spring  for  the  required  relation  of  the 
deflection  to  load  ;  that  is,  it  gives  a  spring  of  the  required  stiff- 
ness, provided  the  stress  produced  does  not  exceed  the  elastic 
limit.  It  is  necessary  to  check  the  spring  as  found  above,  for  if 
the  elastic  stress  is  passed,  the  spring  not  only  takes  a  permanent 
set,  but  the  required  ratio  of  the  load  to  the  deflection  will  not  be 
secured.  On  the  other  hand,  it  is  often  important  for  economy  of 
material  to  use  as  light  a  spring  as  is  consistent  with  safety  ;  or 
in  other  words,  it  is  important  not  to  have  too  low  a  working 
stress  under  the  maximum  load. 

From  eq.  (3)  :— 

/=  -  ?L-  =  3X1000X30X64  =  ,bs 

Abh*  2  X  2.84  X  9 


46 

This  stress  is  beyond  the  elastic  limit  of  any  ordinary  grade  of 
steel,  hence  it  is  probable  that  some  different  form  of  spring  should 
be  used.  A  change  could  be  assumed,  ac  in  the  thickness  of  the 
plate,  and  new  computations  made  with  the  new  data.  A  thinner 
plate  would  reduce  the  stress,  but  it  would  demand  a  wider  spring 
for  the  required  stiffness.  A  more  general  method  will  now  be 
given,  by  which  it  is  possible  to  determine  the  proper  spring  for 
given  requirements  without  the  necessity  of  successive  trial  com- 
putations. 

From  eq.  (3)  :— 


(6) 

^/  ^/ 

From  eq.  (5)  : — 

bk    ~     ~E&~  (7) 

From  eqs.  (6)  and  (7)  :  — 

PLh  —  BPL*  . 
Af  £8 

From  eq.  (3)  : — 

,      D  T  D  T 

(9) 


The  two  equations  (8)  and  (9)  are  in  convenient  form  for  de- 
signing a  flat  spring  when  the  span  (L},  deflection  (8),  load  (/*), 
and  the  material  are  given.  Example  :  The  span  of  a  rectangular 
section  prismatic  flat  spring  (Type  I)  is  30  inches  ;  and  a  load  of 
1000  Ibs.  applied  at  the  middle  is  to  cause  a  deflection  of  1.5 
inches. 

If  the  modulus  of  elasticity  be  30,000,000  and  the  safe  maxi- 
mum working  stress  be  taken  at  50,000  Ibs.  per  sq.  in.,*  required 
the  dimensions  of  the  cross  section,  h  and  b. 


*If  the  spring  is  provided  with  stops  to  prevent  deflection  beyond  a  cer- 
tain amount,  the  stress  due  to  such  deflection  may  be  nearly  equal  to  the 
elastic  limit  of  the  material.  A  very  small  factor  of  safety  is  all  that  is 
necessary. 


47 
From  eq.  (8)  :— 

h=  K*±L  =  lx     50.QOO  X  900  -   =  i  inch. 
E&       6     30,000,000  X  1.5       6 

Taking  //  =  fj  inch,  to  use  a  regular  size  of  stock,/ will  be 
somewhat  less  than  50,000,  or 

/:  50,000  ::  ^V  :  £  ;.'./=  47-ooo. 
From  eq.  (9)  : —  /<?2.^ 

6=  C  ^  =  ^  x  I000^30  X  «5*=^  inches. 

//J2          2  47,000X25 

If  this  width  is  inadmissable,  a  laminated  or  plate  spring  may 
be  used.  See  next  article. 

It  will  be  noted  that  equation  (8)  does  not  directly  involve 
either  the  load  POT  the  breadth  of  spring  b.  It  is  evident  that  if 
a  beam  (flat  spring)  of  given  span  (Z,).  and  thickness  (A),  is 
caused  to  deflect  a  given  amount  (8),  the  outer  fibres  will  undergo 
a  definite  strain  which  is  not  dependent  upon  the  width  of  the 
beam  (d),  nor  upon  the  force  required  to  produce  this  change  in 
relative  positions  of  the  molecules.  As  the  unit  strain  multiplied 
by  the  modulus  of  elasticity  equals  the  unit  stress,  it  follows  that 
this  stress  may  be  computed  from  Z-,  /i.  and  S  (which  determine 
the  strain),  in  connection  with  E  If  the  breadth  of  the  beam 
{b)  is  increased,  the  force  (/*)  required  to  produce  the  given  de- 
flection (8)  will  be  proportionately  increased,  but  the  intensity  of 
stress  is  not  affected  bv  these  changes  alone. 

This  same  conclusion  may  be  reached  from  the  following  rela- 
tion,* in  which  p  =  the  radius  of  curvature  due  to  load. 

'-£-"+$-3 

.•-/=—  (ic) 

2p 

It  appears  from  eq.  (n)  that  the  stress  is  simply  proportional 
to  the  thickness  (h}  and  the  radius  of  curvature  (p),  for  any  given 


*See  Church's  Mechanics,  page  261,  and  Unwin,  page  51. 


48 

value  of  E.  The  span  L,  and  the  deflection  8,  determine  p,  so 
that  eq.  (10)  or  (n)  may  take  the  place  of  eq.  (8).  Equations 
(10)  and  (u)  are  important  in  connection  with  the  design  of 
laminated  springs. 

23.  Laminated,  or  Plate,  Springs  — It  was  shown  in  the 
preceding  article  that  the  maximum  thickness  of  a  simple  flat 
spring  is  fixed  when  the  span,  deflection,  and  modulus  of  elas- 
ticity are  known,  and  the  intensity  of  working  stress  lias  been 
assigned.  [See  eq.  (8).]  With  the  value  of  the  thickness  (//) 
thus  limited  it  will  frequently  happen  that  a, simple  spring  will 
require  excessive  breadth  (b)  to  sustain  the  given  load,  and  it  is 
often  necessary  to  use  a  spring  built  up  of  several  plates  or  leaves. 

Example:  P=i,ooo  Ibs ;  L=T>O";  f  =60,000  Ibs.  per  sq. 
in.;  8  =  2",  and  £"  =  30,000,000.  A  simple  prismatic  spring, 
rectangular  section,  with  load  at  the  middle  of  the  span  (Type  I), 
to  meet  the  above  requirements  would  have  : 

k  =  X/£=lx  60,000x900  inchi 

£8       6      30000,000  x  2 

b=  C P^  =  3  x  -  1°°°  X  3°       =  33/3  inches. 
fh       2      60,000  x  .0225 

This  spring,  consisting  of  a  plate  .15  inch  thick  and  33^/3 
inches  wide,  with  a  span  of  30  inches,  is  evidently  an  impractic- 
able one  for  any  ordinary  case.  Suppose  this  plate  be  split  into 
six  strips  of  equal  width,  each  333-5-6  =  55"  wide,  and  that 
these  strips  are  piled  upon  each  other  as  in  Fig.  19  ;  then,  ex- 
cept for  friction  between  the  various  strips,  the  spring  would  be 
exactly  equivalent,  as  to  stiffness  ;ind  intensity  of  stress,  to  the 
simple  spring  computed  above.  While  the  form  of  laminated 
spring  which  has  just  been  developed  might  answer  in  some 
cases,  another  form,  based  upon  the  "uniform  strength"  beam 
(Type  II),  is  much  better  for  the  ordinary  conditions.  It  may 
be  developed  as  follows,  taking  the  same  data  as  the  preceding 
example  except  that  the  spring  is  to  be  of  Type  II  (Table,  page 
44). 


912 

—  49  — 
In  the  simple  spring,  Type  II, 

h  =  A-/4=-L   x     60-000  X  900  = 
EQ       4         30,000,000  X  2 

b  =  CPL  =  -3.    x      I'000^30-  =  14.8  inches. 
fh          2         6o,oooX.o5o6 

A  huninated  spring  for  the  case  under  consideration  may  be 
derived  from  this  simple  spring  by  imagining  the  lozenge  shaped 
plate  to  be  cut  into  strips  which  are  piled  one  upon  another  as 
indicated  in  Fig.  20.  The  thickness  .225  inches  does  not  corres- 
pond to  a  regular  commercial  size  of  stock,  however,  and  it  will 
usually  be  better  to  modify  the  spring  to  permit  using  standard 
stock.  If  a  thickness  of  ^"  be  assumed  for  the  leaves,  or  plates, 
the  stress,  as  found  from  eq.  (8)  of  the  preceding  article  becomes  : 

/• hE§ 4  X  -25  X  30,000,000  X  2_  ,, 

f-'KL*-  "900—  66'7°°' 

If  this  stress  is  considered  too  great,  we  might  use  steel  T3g-" 
thick,  when  /=  1  X  3  X  3o.ooo,ooo_X  2  = 
16  X  900 

With  h  —  ^",  and  f=-  50,000, 

b  =  cPL        3    x  LOOP  X3Q>056  =  25.6". 
ft?        2  50,000  X  9 

If  this  spring,  30"  span,  fV  thick,  and  25.6"  wide  at  the 
middle,  be  replaced  by  5  equivalent  strips,  each  25.6  H-  5  =  5.11" 
wide  (nearly  5^"),  see  Fig.  20,  a  laminated  spring  of  good  form 
and  practicable  dimensions  will  result.  In  cases  where  the  maxi- 
mum allowable  width  of  spring  is  fixed,  a  larger  number  of 
plates  may  be  necessary.  Thus,  in  the  preceding  problem,  if  the 
spring  width  must  be  kept  within  4/^",  it  is  necessary  to  use  6 
plates,  each  25.6  -i-  6  =  4.27"  wide.  In  actual  springs,  the  usual 
construction  is  that  shown  by  Fig.  21,  in  which  the  several  plates 
have  the  ends  cut  square  across  instead  of  terminating  in  tri- 
angles. These  springs  approximate  uniform  strength  beams,  and 


—  5o  — 

may  be  computed  by  equations  (8)  and  (9)  of  art.  22,  remember- 
ing that  b  is  the  breadth  of  the  equivalent  simple  spring.  Or,  if 
n  is  the  number  of  plates  and  b\  the  breadth  of  each  plate  in  the 
laminated  spring,  nbl  =  b. 

The  last  of  these  formulas,  eq.  (9),  is  not  strictly  applicable 
when  the  ends  of  the  plates  are  cut  square  across  ;  but  it  may 
generally  be  used  with  sufficient  accuracy,  provided  the  succes- 
sive plates  are  regularly  shortened  by  uniform  amounts.  It  is 
quite  common  practice  to  have  two  or  more  of  the  plates  extend 
the  full  length  of  the  spring.  This  construction  makes  the 
spring  a  combination  of  the  triangular  and  prismatic  types  ; 
(Type  II  and  Type  I,  or  Type  V  and  Type  IV,  depending  upon 
whether  the  spring  is  supported  at  the  ends,  or  is  a  cantilever). 
Mr.  G.  R.  Henderson  in  discussing  the  cantilever  form,  (Trans. 
A.  S.  M.  E.  vol.  XVI)  says  :—  "  For  a  spring  with  all  the  plates 
full  length  we  would  have 


EnbJ? 

so  for  one-fourth  of  the  leaves  full   length,   the  deflection  would 
be  decreased  approximately  one-fourth  of  the  difference  between 


or 


Enbfi         Enbh*  Enbh*  ' 

By  similar  reasoning,  for  a  spring  Loaded  at  the  middle  and 
supported  at  the  ends,  with  one-fourth  the  plates  extending  the 
whole  length  of  the  spring, 


32  Enbjt  ' 

This  may  be  otherwise  stated  as  follows  : 

When  the  number  of  full  length  leaves  is  one-fourth  the  total 
number  of  leaves  in  the  spring,  use  f^  B  instead  of  B  and  \\  K 
instead  of  K  in  equations  (5)  and  (8)  of  the  preceding  article  ; 
the  value  of  K  being  that  given  for  the  triangular  forms,  Type 
II  or  Type  V,  as  the  case  may  be. 


The  spring  shown  in  Fig.  21  is  initially  curved  (when  free), 
which  is  common  practice.  The  best  results  are  obtained  by 
having  the  plates  straight  when  the  spring  is  under  its  normal 
full  load  (if  this  is  practicable)  because  the  sliding  of  the  plates 
upon  each  other,  with  the  vibrations,  is  then  reduced  to  a  mini- 
mum. 

The  several  plates  of  a  laminated  spring  are  usually  secured  by 
a  band  shrunk  around  them  at  the  middle  of  the  span.  This 
band  stiffens  the  spring  at  the  middle,  and  ^  the  length  of  the 
band  (y2 \  I,  Fig.  21)  may  be  deducted  from  the  full  span  to  give 
the  effective  span  to  be  used  as  L  in  the  above  formulas.  It  is 
not  uncommon  to  make  the  longest  plate  thicker  than  the  others, 
if  but  one  plate  is  given  the  full  length  of  the  spring.  This  can- 
not be  looked  upon  as  desirable  practice,  however,  as  all  of  the 
plates  are  subjected  to  the  same  change  in  radius  of  curvature  ; 
hence  the  thicker  plate  is  subjected  to  the  greater  stress.  See 
eq.  (i  i),  art.  22. 

The  following  formulas  (derived  from  the  preceding)  may  be 
used  in  computing  flat  springs  ;  but  it  must  be  remembered  that 
there  is  always  liability  of  considerable  variation  in  the  modulus 
of  elasticity,  hence  such  computations  can  only  be  expected  to 
give  approximations  to  the  deflections  which  will  be  observed  by 
tests  of  actual  springs.  These  computations  will  be  sufficiently 
exact  for  m  my  purposes  ;  but  when  it  is  important  to  accurately 
determine  the  scale  of  the  spring  (ratio  of  deflection  to  load), 
actual  tests  must  be  made.  In  using  these  formulas  the  following 
rules  should  be  observed. 

I.  When   the  several   plates  are  secured  by  a  band  shrunk,  or 
forced,  over  them,  one-half  the  length  of  the  band  is  to  be  sub- 
tracted from  the  length  of  the  spring  to  get  the  effective  length  of 
the  spring. 

II.  When    the   plates    have   different    thicknesses,    the   stress 
should  be  computed   from  the  plate  having  the  maximum  thick- 
ness. 

III.  If  more  than  one  plate  has  the  full  length  of  the  spring, 
an  appropriate  modification  of  the  values  of  the  coefficients  B  and 


—  52  — 

and  K  should  be  made.  Thus,  when  one-fourth  of  the  total 
number  of  plates  are  full  length,  \\  B  and  \\  K  should  be  used 
instead  of  B  and  A^(Type  II  or  V)  in  equations  I,  II,  III,  and  IV. 

EQUATIONS. 


A  B.-~K-  (III) 


/=  £L?  (V) 

p=AfnblV  (VI) 

JL> 


/>/ 

b  =  -----    =    L±_~  (VIII) 

^  «/As        «/A2 

Kxperience  shows  that  thin  plates  have  a  higher  elastic  limit 
than  thick  plates  of  similar  grade  of  material.  In  the  practice  of 
a  prominent  eastern  railway  company,  the  values  allowed  for  the 
maximum  intensity  of  stress  in  flat  steel  springs  are,  for  : 

Plates  ^  inch  thick,  ^=90,000  Ibs.  sq.  in. 
"      fV     "         "       7=84,000    " 
"        f  "       /^So.ooo    " 

"      A     "        "      /=77.ooo    " 
\  /=75,ooo    ' 

The  above  values  are  satisfied  by  the  equation  /=  60,000  + 
Zl3 —  ,  in  which  h  is  the  thickness  of  plate. 


—  53  — 

These  values  are  for  the  greatest  stress  to  which  the  material 
can  be  subjected,  as  when  the  spring  is  deflected  down  against 
the  stops 

The  modulus  of  elasticity,  £,  may  vary  considerably  ;  but  its 
value  may  be  assumed  at  about  30,000,000  in  the  absence  of  more 
definite  data. 

In  designing  a  new  spring,  the  value  of  h  is  to  be  found  from 
equation  (III)  ;  then  b  is  found  by  equation  (VIII).  The  other 
formulas  are  useful  in  checking  springs  already  constructed  for 
deflection  due  to  a  given  load,  or  the  reverse  ;  for  safety,  etc. 

24.  Helical  Springs. — [Unwin,  §343,  pages  72,  73.]  If  a  rod 
or  wire  be  wound  into  a  flat  ring  with  the  ends  bent  in  to  the  cen- 
tre, Fig  22,  and  two  equal  and  opposite  forces,  +  P  and  —  P,  be 
applied  to  these  ends  (perpendicular  to  the  plane  of  the  ring)  as 
indicated,  the  rod  will  be  subjected  to  torsion. 

If  a  longer  rod  be  wound  into  a  helix,  with  the  two  ends  turned 
in  radially  to  the  axis,  the  typical  helical  spring  is  produced.  If 
two  equal  and  opposite  forces,  +  P  and  —  P,  act  on  these  ends, 
along  the  axis  of  the  helix,  they  induce  a  similar  stress  (torsion) 
in  the  rod,  but  as  the  coils  do  not  lie  in  planes  perpendicular  to 
the  line  of  the  forces,  there  is  a  component  of  direct  stress  along 
the  rod.  This  direct  stress  increases  as  the  pitch  of  the  coils  in- 
creases relative  to  their  diameter  ;  but  with  ordinary  proportions 
of  springs,  the  torsion  alone  need  be  considered,  when  the  exter- 
nal forces  lie  along  the  axis  of  the  helix. 

The  following  notation  will  be  used  in  treating  of  helical 
springs  of  circular  wire,  subjected  to  an  axial  load  : 

P  =  the  force  acting  along  the  axis. 

r  =  the  radius  of  the  coils,  to  centre  of  wire. 

d  =  the  diameter  of  wire. 
f  =  the  maximum  intensity  of  stress  in  wire  (torsion). 

/p  =  the  polar  moment  of  inertia  of  wire. 

G  =  the  transverse  modulus  of  elasticity. 

8    =  the  "deflection  "  (elongation  or  shortening)  of  spring. 

^  =  the  number  of  coils  in  the  spring. 

L  =  the  length  of  wire  in  the  helix  =  2  TT  r  >i  (approximately). 


—  54  — 

Suppose  a  helical  spring  under  an  axial  load  to  be  cut  across 
the  wire  at  any  section,  and  the  portion  on  one  side  of  this  section 
to  be  considered  as  a  free  body,  Fig.  23.  Neglecting  the  direct 
stress,  equilibrium  demands  that  the  moment  of  the  external  force 
(Pr)  shall  equal  the  stress  couple,  or  moment  of  resistance, 

(v-  fd*  for  circular  section   \. 
16 

If  this  free  portion  of  the  helix  is  straightened  out.  as  indicated 
by  the  broken  lines  in  Fig.  23,  till  its  direction  is  perpendicular 
to  the  radial  end,  it  will  appear  that  the  moment  Pr  still  equals 

the  moment  of  resistance,  --  fd*.     Since  the  stress  and  strain  are 

the  same  in  this  helix  and  the  straight  rod.  it  appears  that  the 
energy  expended  against  the  resilience  is  the  same  in  both  cases 
(the  length  of  wire  affected  remaining  constant).  Or,  as  the  force 
(P)  and  the  arm  (r~)  are  the  same  in  both  conditions,  the  distances 
through  which  this  force  acts  to  produce  a  given  torsional  stress 
(f)  are  equal.  If  a  straight  rod  of  length  L  is  subjected  to  a  tor- 
sion moment  Pr,  the  angle  of  twist  being  a  (in  TT  measure), 


[See  Church's  Mechanics,  page  236.] 

The  energy  expended  on  the  rod  is  the  mean  force  applied  mul- 
tiplied by  the  distance  through  which  this  force  acts  If  the  load 
is  gradually  applied,  this  energy  is  \Pra.  In  the  case  of  the 
corresponding  helical  spring,  the  mean  force  Of/*)  acts  through  a 
distance  equal  to  the  ' '  deflection  ' '  of  the  spring  (8),  or  the  energy 
expended  is  \  P8.  As  pointed  out  above,  the  energy  expended 
in  the  two  cases  is  the  same,  or 

Y 

Pr  ^  a /p  G  _  8       TT d4          G     =  ^(T_G 
L  r         32        2-irrn       64^71 

(i) 


—  55  — 

Equation  (i)  mav  be  used  for  finding  the  load  corresponding  to 
an  assigned  deflection  in  a  given  spring.  The  equation  can  be 
put  in  the  following  form  for  finding  the  deflection  due  to  a  given 
load: 


Or  the  equation  may  be  employed  for  designing  a  spring  in  which 
the  load  and  deflection  are  given,  by  assuming  any  two  of  the 
three  quantities,  r,  (/and  n.  The  most  convenient  form  for  this 
latter  purpose  is  usually, 

(3) 


64  Pr3 

These  equations  for  rigidity  only  hold  good  within  the  elastic 
limit  of  the  material,  as  G  is  simply  a  ratio  between  stress  and 
strain  within  this  limit.  It  therefore  becomes  necessary  to  check 
any  of  the  above  indicated  computations  for  strength,  and  it  will 
often  be  found,  after  thus  checking,  that  the  stress  is  either  too 
high  for  safety,  or  too  low  for  economy. 

The  formula  for  the  strength  of  a  solid  circular-section  rod  un- 
der torsion  is 

(4) 
f-  ^Pr 

J  J3 


I6P    ' 

It  is  to  be  remembered  that  as  equation  (4)  is  for  safe  strength, 
the  load  (/*)  should  be  the  maximum  load  to  which  the  spring 
can  be  subjected  ;  but  equation  (3)  may  be  used  with  any  load 
and  the  corresponding  deflection. 

Example:  The  load  on  a  helical  spring  is  1600  Ibs.,  and  the 
corresponding  deflection  is  to  be  4".  Transverse  modulus  of  elas- 
ticity of  material  =  11,000.000,  and  the  maximum  intensity  of 
safe  torsional  stress  —  60,000  Ibs.  wire  of  circular  section.  To 
design  the  spring,  assume  d  =  f",  and  r—  i£"  ;  from  eq.  (3), 


-56- 

n  __  4  X  625  X  11,000.000  X  8       i 
4096  X  64  X   1600  X  27 

Checking  for  the  stress  by  the  last  equation  in  group  (4), 
/=  16  X  i^oo  X  i,5_X  _5i2  =  lbs 


I25 


This  stress  is  found  to  be  safe,  but  is  considerably  below  the 
limit  assigned,  and  it  may  be  desirable  to  work  up  to  a  somewhat 
higher  stress.  Another  computation  can  be  made  (with  a  smaller 
d  or  larger  r),  and  by  a  series  of  trials,  the  desired  spring  can 
be  found.  The  following  order  of  procedure  avoids  this  element 
of  uncertainty.  The  load  being  given,  assume  a  diameter  of  wire 
and  value  of  safe  stress,  then  solve  in  eq.  (4)  for  the  radius  of  coil. 
Make  this  radius  some  convenient  dimensions  (not  exceeding  that 
computed  if  the  assumed  stress  is  considered  the  maximum  safe 
value).  Next  substitute  these  values  of  d  and  r  (with  those 
given  for  P,  8  and  G)  in  eq.  (3)  to  find  the  number  of  coils.  Thus, 
with  the  data  of  the  preceding  example,  assuming  d  =  f"  ; 

r  =  v  fd*  —  ^X  60,000  X  125  _          /, 
16    P  ~  16  X  1600  X  512 

If  the  f"  rod  is  wound  on  an  arbor  3"  diameter,  the  radius  to  the 
centre  of  coils  will  be  about  1.81"  ;  and  the  corresponding  stress 
would  be  60,500  lbs.  per  square  inch.  This  is  so  slightly  in  ex- 
cess of  the  assigned  value  that  it  may  be  permitted,  especially  as 
this  value  is  a  moderate  one  for  spring  steel.  Substituting  in 

eq-  (3), 

__  od^G  _     4  X  11,000,000  X  625 

~~  64 /V  ~~  64  X  1600  X  5.93  X  4096 

It  may  be  desirable  to  fix  upon  the  radius  of  coil,  rather  than  the 
diameter  of  wire,  in  the  first  computation,  in  designing  a  spring. 
From  eq.  (4) : 

(5) 


—  57- 

In  other  cases,  it  may  be  desirable  to  assume  the  ratio  of  the  radi- 
us of  coil  to  the  diameter  of  wire,  then  from  eq.  (4)  : 


(6) 

In  either  of  the  preceding  conditions,  use  a  regular  size  of  wire. 

In  checking  a  given  spring,  it  may  be  required  to  determine 
either  the  safe  load,  or  the  safe  deflection.  If  the  former  is  the 
case,  eq.  (4)  may  be  used  directly.  If  it  is  required  to  find  the 
safe  deflection,  substitute  the  value  of  P  from  eq.  (4)  in  eq  (2) 
and  the  result  is 

»="•**/'  (7) 

The  weight  of  a  spring  is  a  matter  of  some  importance,  as  the 
material  is  expensive.  The  following  discussion  shows  that  the 
weight  varies  directly  as  the  product  of  the  load  and  the  deflec- 
tion, inversely  as  the  square  of  the  intensity  of  stress  in  the  wire, 
and  directly  as  the  transverse  modulus  of  elasticity.  Hence  for 
a  given  load  and  deflection,  economy  calls  for  a  high  working 
stress  and  a  low  modulus  of  elasticity.  From  eq.  (4)  : 

P  =•    ,./  —  !   also  for  a  member  under  torsion, 
16      r 

f  =•        '  —.-     [Church's  Mechanics,  p.  235]. 


(8) 


But  the  volume  of  the  spring  is 

(10) 


f-d  v8  v       G 

J   —     —   /\  —  A  -  -  --  -  — 

2        r       2-jrrn      ^irr  n 


-58- 


(ii) 


The  weight  is  directly  proportional  to  the  volume  ;  hence,  for 
given  values  of  G  and  /,  the  weight  varies  simply  as  the  product 
of  the  load  and  the  deflection.  All  possible  helical  springs  (of 
similar  section  of  wire)  have  the  same  weight  for  a  given  load  and 
deflection,  if  of  the  same  material  and  worked  to  the  same  stress. 
It  can  be  shown  that  a  helical  spring  of  square  wire  must  have  50 
per  cent,  greater  volume  than  one  of  round  wire,  the  stress  and 
modulus  of  elasticity  being  the  same  in  both.  The  round  section 
is  generally  admitted  to  be  best  for  helical  springs  under  ordinary 
conditions. 

A  small  wire  of  any  given  steel  usually  has  a  higher  elastic 
limit  than  a  larger  one,  while  there  is  not  a  corresponding  change 
in  the  modulus  of  elasticity  with  change  in  diameter.  This  sug- 
gests the  use  of  as  light  a  wire  as  is  consistent  with  other  require- 
ments. 

An  extensive  set  of  tests  of  springs,  conducted  by  Mr.  E.  T. 
Adams,  in  the  Sibley  College  Laboratories,  indicate  that  the  steel 
such  as  is  used  in  governor  springs  may  be  subjected  to  stress 
varying  from  about  60,000  Ibs.  per  square  inch  with  f  "  wire  to 
80,000  Ibs.  per  square  inch  (or  more)  in  wire  f  "  diameter.  The 
following  expression  may  be  used  to  find  the  safe  stress  in  such 
springs  : 

/-  40,000  +'5^?.  (12) 

a 

Mr.  J.  W.  Cloud  presented  a  most  valuable  paper  on  Helical 
Springs  before  the  Am.  Society  of  Mechanical  Engineers  (Trans. 
Vol.  V,  page  173),  in  which  he  shows  that  for  rods  used  in  rail- 
way springs  (f"  to  if\"  diam.)  the  stress  may  be  as  high  as  80,000 
Ibs.  per  square  inch,  and  that  the  transverse  modulus  of  elasticity 
is  about  12,600,000. 

Two  or  more  helical  springs  are  often  used  in  a  concentric  nest 
(the  smaller  inside  the  larger)  ;  all  being  subjected  to  the  same 
deflection.  This  is  common  practice  in  railway  trucks,  where  the 
springs  are  under  compression  when  loaded.  If  these  springs 
have  the  same  "  free  "  heighth  (when  not  loaded),  and  if  they  are 


—  59  — 

of  equal  heighth  when  closed  down  "solid,"  Mr.  Cloud  shows 
that  the  length  of  wire  should  be  the  same  in  each  spring  of  the 
set  for  equal  intensity  of  stress.  The  ' '  solid  ' '  heighth  of  a  spring 
is  // -  -  d  n,  and  the  length  of  wire  is  L  =  2  TT  r  n  ;  hence  the  num- 
bers of  coils  of  the  separate  springs  of  the  set  are  inversely  as  the 
diameters  of  the  wire  and  inversely  as  the  radii  of  the  coils  ;  or 
the  ratio  of  r  to  d  is  the  same  in  each  spring  of  the  nest.  This 
conclusion  may  be  somewhat  modified  when  it  is  remembered 
that  the  wire  of  smaller  diameter  may  usually  be  subjected  to 
somewhat  higher  working  stress  than  the  larger  wire  of  the  outer 
helices  ;  and  also  that  the  wire  of  these  compression  springs  is 
commonly  flattened  at  the  end  to  secure  a  better  bearing  against 
the  seats.  See  Fig.  24. 

Two  common  methods  of  attaching  "  pull  "  springs  are  shown 
in  Fig.  25.  One  end  of  the  spring  shows  a  plug  with  a  screw 
thread  to  fit  the  wire  of  the  spring.  This  plug  is  usually  tapered 
slightly,  and  the  coils  of  the  spring  are  somewhat  enlarged  by 
screwing  it  in.  The  other  end  of  the  spring  shows  the  wire  bent 
inward  to  a  hook  which  lies  along  the  axis  of  the  helix.  The 
former  method  is  usually  preferable  for  heavy  springs. 

SUMMARY   OF   HELICAL  SPRING   FORMULAS. 

Pr=  ~  fd:'  (I)        8  =  I2^57  "//  (vn) 

16  Gd 

r="fd*  (II)  P  •-•=¥?£ 

04  t  n 
^  (III)        8=64/Vn  (IX) 


(V)         v=P*  (XI) 

2  Cr 


«  (VI) 

16   r 


—  6o  — 

Formulas  (I)  to  (VII),  inclusive,  relate  to  strength  ;  (VIII)  to 
(X),  inclusive,  relate  to  rigidity,  or  elasticity. 

In  the  absence  of  more  exact  information  as  to  the  properties  of 
the  material  of  which  a  steel  helical  spring  is  made,  the  following 
values  may  be  taken  : 

G  =  12,000,000, 

/=  40, ooo  4-  I5'°°5>. 
a 

25.  Spiral  or  Helical  Springs  in  Torsion  — The  following 
formulas  for  either  true  spiral  or  helical  springs  subjected  to  tor- 
sion are  derived  from  "The  Constructor,"  by  Professor  Reu- 
leaux. 

PRL        ,      WR 


In  which 


P  =  load  applied  to  rotate  axle, 

R  =  lever  arm  of  this  load, 

<£  =  angle  through  which  axle  turns, 

L  =  length  of  effective  coils, 

E=  modulus  of  elasticity  (direct), 

/=  moment  of  inertia  of  the  section. 


IV 
PIPES,  TUBES  AND  FLUES. 


26.  Pipes,  Tubes  and  Flues,  of  wrought  iron,  steel  and  cast 
iron  have  many  applications  in  mechanical  constructions,  and 
the  requirements  are  quite  different  in  various  services.  How- 
ever, there  are  certain  standard  forms  well  adapted  to  varied 
uses . 

Wrought  iron  or  steel  pipes,  such  as  are  used  for  steam,  water, 
gas,  etc.,  are  designated  by  the  nominal  inside  diameters  ;  while 
tubes,  such  as  are  used  in  boilers,  are  rated  by  the  outside  diame- 
ters. The  process  of  making  ordinary  pipes  and  tubes  is  to  roll 
a  long  strip  into  a  tube  somewhat  larger  in  diameter  than  the 
finished  size  ;  then  to  draw  this  tube,  while  at  a  welding  heat, 
through  reducing  dies.  This  welds  the  edges  together  and 
reduces  the  pipe  (or  tube)  to  the  required  size.  Small  size  pipes 
(usually  up  to  i  inch)  are  "  butt  welded  "  ;  larger  sizes  are  "  lap 
welded." 

There  are  numerous  processes  for  making  seamless  tubes. 
These  tubes  are  largely  used  for  bicycle  frames,  and  to  some 
extent  for  other  service.  They  are  much  stronger  than  welded 
tubes,  as  the  latter  usually  fail,  if  at  all,  by  splitting  along  the 
seam.  The  welded  tube  is  much  cheaper,  however,  and  is  safely 
used  for  most  services. 

Boiler  tubes  are  thinner  than  pipes  of  similar  diameter,  because 
the  latter  are  given  sufficient  thickness  to  permit  cutting  threads 
on  the  ends  Owing  to  the  process  of  manufacture,  the  outside 
diameter  varies  but  little  from  the  standard  size,  while  any  varia- 
tion in  thickness  of  metal  produces  variation  of  inside  diameter. 
A  two  inch  tube  will  be  very  nearly  2"  outside  diameter,  while 
an  ordinary  2"  pipe  is  about  2$"  outside  diameter  and  2^"  i"- 
side  diameter.  Pipes  usually  have  an  actual  inside  diameter 
rather  greater  than  the  nominal  size.  This  variation  exceeds  one- 


—  62  — 

eighth  of  an  inch  in  certain  sizes  ;  while  in  a  standard  2^"  pipe 
(and  some  few  of  the  large  sizes)  the  actual  inside  diameter  is 
slightly  less  than  the  nominal  dimension.  Beside  the  standard 
pipes,  which  have  an  ample  factor  of  safety  against  bursting 
under  ordinary  pressures,  there  are  thicker  pipes  known  as 
"extra  strong",  and  "  double  extra  strong."  These  latter  are 
suitable  for  high  pressures,  as  in  connection  with  hydraulic  ma- 
chinery, etc.  The  "  extra  strong  "  and  "  double  extra  strong  " 
pipes  are  drawn  by  dies  the  same  diameter  as  those  used  for  the 
same  nominal  sizes  of  standard  pipes  ;  hence  the  inside  diameters 
are  considerably  less,  owing  to  the  extra  thickness  of  walls. 
The  following  actual  dimensions  illustrate  : 

TWO    INCH    PIPE. 

Outside  Diam.        Inside  Diam. 

Standard,  2.375"  2.067" 

Extra  Strong,  2.375  J-933 

Double  Extra  Strong,     -  2-375  l<491 

Various  books  and  catalogues  contain  tables  of  dimensions  of 
pipes  and  tubes. 

Wrought  iron  and  steel  pipes  are  usually  joined  together  by 
screwing  the  threaded  ends  into  couplings  (sleeves),  or  into 
flanges.  The  former  method  is  most  common  in  sizes  up  to 
about  6",  while  bolted  flanges  are  commonly  used  with  larger 
sizes. 

The  Briggs  Standard  Pipe  Threads,  almost  universally  used  in 
this  country,  have  for 

YZ"  pipe,  27  threads  per  inch. 
%"  and  3/%"  pipe,  18  threads  per  inch. 
YZ"  and  y^"  pipe,  14  threads  per  inch, 
i"  to    2"  pipe,  11^2  threads  per  inch. 
2^"  and  over,  8  threads  per  inch. 

For  form  of  threads  and  other  details  as  to  Briggs  system,  see 
Trans.  A.  S.  M.  E.,  Vol.  VIII,  page  29. 

There  is  much  variation  in  the  practice  of  various  makers  of 
fittings  in  the  dimensions  of  pipe  flanges.  A  committee  of  the 


-63- 

American  Society  of  Mechanical  Engineers  suggested  a  standard 
(Trans.,  Vol.  XIV,  page  49),  after  considering  the  various  re- 
quirements and  existing  practice.  This  system  is  by  no  means 
generally  adopted  as  yet,  however. 

Cast  iron  pipes  are  generally  used  for  water  works  systems. 
These  are  sometimes  joined  by  flanges  cast  with  the  sections  of 
pipe,  but  more  frequently  the  joint  is  of  the  "bell  and  spigot" 
form.  The  "bell"  is  a  cup  shaped  enlargement  on  one  end  of 
the  section,  into  which  the  smaller  end  of  the  adjacent  pipe  is  in- 
serted. After  placing  the  "spigot"  in  the  "bell"  the  annular 
space  is  calked  with  "oikuin"  or  other  fibrous  material;  then 
melted  lead  is  poured  in,  and  afterwards  calked,  to  retain  the 
soft  packing.  The  spigot  end  is  usually  provided  with  a 
"  bead  "  to  make  it  less  liable  to  work  out  This  form  of  joint 
permits  a  certain  degree  of  flexibility,  not  secured  with  bolted 
flanges,  which  is  desirable  if  the  ground  settles  unequally  under 
the  pipe. 

Cast  iron  pipe  should  be  cast  on  end  (with  the  axis  vertical)  as 
this  reduces  the  danger  of  producing  an  unsymmetrical  pipe 
through  springing  of  the  core,  or  accumulation  of  dirt  on  one 
side  of  the  casting. 

Tubes  are  used  in  boilers  either  for  conveying  the  hot  gases 
through  the  water,  (fire  tubes),  or  for  conducting  the  water 
through  the  hot  gases  (water  tubes).  Large  tubular  passages 
for  the  hot  gases  are  called  flues.  These  latter  very  often  have 
riveted  seams  instead  of  welded  ones. 

If  a  pipe  is  used  for  conveying  water  (or  other  fluid)  under  heavy 
pressure,  the  primary  straining  action  is  a  bursting  one,  which 
produces  a  tensile  stress  in  the  material.  There  is  also  liability 
of  considerable  shock  ("  water  hammer  ")  in  many  cases.  With 
gases  or  dry  vapors  there  is  less  liability  of  shock  due  to  sudden 
change  in  the  velocity  of  the  flow  than  with  liquids  ;  but  faulty 
drainage  with  pipes  containing  steam  (or  other  vapors  which 
condense  readily)  may  result  in  most  violent  shock  to  the  line. 
In  gas  mains,  exhaust  steam  pipes,  etc.,  the  bursting  pressure  is 
often  small.  These  lines  may  be  in  greater  danger  from  such 


-64- 

irregular  actions  as  unequal  settling  of  the  ground  (if  buried),  or 
of  the  supports  if  carried  otherwise  ;  from  temperature  changes, 
etc.  Expansion  and  contraction  frequentl}7  result  in  most 
dangerous  straining  actions,  unless  properly  provided  for.  Boiler 
tubes  and  flues  may  be  permanently  injured,  or  so  temporarily 
weakened  as  to  cause  initial  rupture,  by  becoming  over  heated 
through  low  water.  Some  of  the  above  actions,  for  example  ex- 
pansion and  contraction,  may  often  be  foreseen  and  proper  pro- 
vision be  made  for  them  ;  while  such  straining  actions  a-;  unequal 
settling  of  supports,  over-heating,  etc.,  are  less  determinate. 

27.  Resistance  of  Thin  Cylinders  to  Internal  Pressure. 
[Unwin,  §  26a,  page  47  ]  If  a  cylinder  of  circular  section,  sub- 
jected to  an  internal  unit  bursting  pressure  (/>),  has  a  thickness 
(/)  which  is  small  relative  to  its  diameter  (d),  the  intensity  of 

stress  along  any  longitudinal  section  is  /"=<__,    as  given  by  Un- 

win. Fig-  26  shows  one  half  of  such  a  cylinder  as  a  free  body. 
The  normal  pressure  on  a  longitudinal  strip  of  length  /  and 
width  rdd  is  plrdd.  The  total  stress  on  the  two  free  sections 
(each  of  area  =  //)  is 

2/J=2///=    C  plrdOs\\\  Q^filr  C   sin  Od6=iplr  =^  p  d  I 

Jo  Jo 

*  .-./=£  (.) 

>=*-  (,) 


'= 

If  a  transverse  section  of  the  cylinder  be  considered,  (Fig.  27), 
it  will  be  seen  that  the  total  pressure  on  the  head,  which  tends 

*The  result  would  have  been  the  same  if  the  length  of  the  shell  had  been 
treated  as  unity,  because  the  total  stress  and  the  area  over  which  this  stress 
is  distributed  both  vary  directly  as  the  length,  or  /  cancels  out  in  any  case. 


to  cause  rupture  along  a  transverse  section,  is   —  d'L  p  ;   and    this 

4 

is  equal  to  the  intensity  of  stress  produced  multiplied  by  the  area 
of  metal  in  such  a  section  or 

*  dlp=7rdtf 

,.  4/=ff  (4) 

/  =  (5) 


A  comparison  of  (r)  and  (4)  shows  the.  stress  in  transverse  sec- 
tions to  be  only  one-half  that  in  longitudinal  sections.  For 
this  reason  it  is  very  common  practice  to  make  circumferential 
seams  of  a  boiler  shell  single  riveted,  when  the  longitudinal 
seams  are  double  riveted. 

A  comparison  of  (2)  and  (5)  shows  that  for  a  sphere  (all  of  the 
sections  of  which  correspond  to  transverse  sections  of  a  cylinder) 
the  pressure  is  twice  as  great  as  in  a  cylinder  of  the  same  thick- 
ness and  diameter  for  the  same  maximum  stress. 

In  a  cylinder  (pipe  or  flue)  which  has  a  riveted  seam,  the  com- 
putations should,  of  course,  be  made  for  a  section  passing 
through  such  seam.  The  ratio  of  the  strength  of  the  section 
through  the  seam  to  the  strength  of  a  parallel  section  through 
the  solid  plate  is  called  the  "efficiency  of  the  joint"  (e).  The 
method  of  computing  the  efficiency  for  any  form  of  riveted  joint 
will  be  given  in  the  next  chapter  ;  but  it  is  always  less  than 
unity  and  may,  when  known,  be  used  as  follows  : 

The  stress  in  the  solid  plate  equals  the  stress  at  the  joint  multi- 
plied by  the  efficiency  ;  then  if  f  is  the  stress  allowed  at  the 
weakened  sections,  e/  is  the  stress  in  the  solid  plate  or,  for  the 
longitudinal  sections, 


—  66  — 


'="'' 


28.  Resistance  of  Non-Circular  thin  Cylinder  to  Internal 
Pressure.  —  Suppose  a  cylinder  to  have  a  cross-section  made  up 
of  circular  arcs,  as  in  Fig.  28.  Take  the  upper  half  as  a  free 
body  (section  along  the  major  axis).  L,et  the  resultants  of  the 
components  of  pressure  which  are  normal  to  the  plane  of  the  sec- 
tion be  />,,  P.2,  and  P3  for  the  portions  marked  I,  II,  and  III,  re- 
spectively. Then  these  resultant  forces,  per  unit  of  length  of  the 
cylinder,  are  as  follows. 

/•*' 
/*,  =P  r  I      sin  <£  d  <£  =p  r  (  —  cos  <£'  +  cos  o)  =  p  ml  ; 

J  o 

r*' 
/>2  =  p  R  I      sin  6  d  0  =^p  R  (—  cos  0'  -f  cos  6")  =  p  m,  ; 

*/  8/  '  ^ 

Ps  =  t>  r  I        sin  <£  d  <^>  =  p  r  (—  cos  -n-  +  cos  ^>")  =  p  m.A. 
J  y 

:.  P,  +  P,  +  P,  =  p  (m,  +  m,  +  m,  )  =  p  ^. 
In  a  similar  way,  if  the  section  is  taken  along  the  minor  axis, 
the  resultant  force  normal  to  this  axis  is  found  to  be  p  B  .  In  like 
manner  the  resultant  force  normal  to  any  section  is  (per  unit  of 
length  of  cylinder)  equal  to  the  intensity  of  pressure  multiplied 
by  the  axis  of  that  section.  As  B  is  less  than  ^f,  the  resultant 
force  p  B  is  less  than  pA  ;  or  the  force  tending  to  elongate  the 
minor  axis  is  greater  than  the  force  tending  to  elongate  the  major 
axis.  If  the  tube  were  perfectly  flexible,  its  form  of  cross-section 
would  become,  under  pressure,  one  in  which  all  axes  are  equal, 


Fig.  42. 


-67- 

or  circular.  A  rigid  material  offers  resistance  to  such  change  of 
form  ;  and  a  flexunil  stress  is  produced  in  addition  to  the  direct 
tension,  but  it  approaches  nearer  to  the  circular  form  as  the  press- 
ure increases.  The  existence  of  this  flexure  stress  in  a  non- 
circular  cylinder  becomes  apparent  from  a  comparison  of  Figs. 
29  and  30  In  Fig.  29  (circular  section)  the  lines  of  normal 
pressure  all  pass  through  a  single  point  (the  centre  of  the  circle)  ; 
the  resultant  (Pr)  of  the  tensions  (/*,  and  P2}  also  passes  through 
this  same  point,  hence,  these  forces  form  a  concurrent  system, 
and  they  are  in  equilibrium.  In  Fig.  30,  however,  the  pressures 
do  not  in  themselves  form  a  concurrent,  nor  parallel,  system  of 
forces,  hence,  they  cannot  be  balanced  by  a  single  force  (as  the 
resultant  Pr),  but  there  must  be  a  moment,  or  moments,  of  stress 
for  equilibrium.  A  similar  course  of  reasoning  could  be  applied 
to  a  cylinder  of  any  non-circular  cross  section  ;  for  such  a  section 
(Fig.  31)  could  be  considered  as  made  up  of  circular  arcs,  each  of 
which  could  be  treated  (like  the  special  case  of  Fig.  28)  by  inte- 
grating between  proper  limits.  A  direct  inspection  will  also 
show  that  in  any  non-circular  section  cylinder,  subjected  to  in- 
ternal pressure,  the  pressure  tends  to  reduce  the  cylinder  to  a 
circular  cross-section.  Suppose  the  original  cylinder  (Fig.  31)  to 
be  cut  along  the  greatest  axis  of  its  cross-section,  and  that  a  flat 
bottom,  coinciding  with  this  section-plane  be  secured  to  it,  as  in 
Fig.  32.  The  total  pressure  on  this  bottom  evidently  balances 
the  components  of  the  pressure  on  the  curved  surface  which  lie 
normally  to  this  flat  bottom  ;  hence,  the  resultant  of  these  normal 
components  of  pressure  equals  p  (a  .  .  a)=pA,  per  unit  of 
length  of  cylinder.  In  a  similar  way,  the  resultant  of  compon- 
ents of  pre>stire  acting  normally  to  any  other  section,  (as  b  .  .  b, 
Fig.  31)  equals  p  (b  .  .  b)  —  pB  <ip  A.  This  direct  method 
might  have  been  used  in  the  preceding  cases  (Figs.  26  and  28) 
without  recourse  to  the  calculus. 

It  is  apparent,  then,  that  any  cylinder  under  internal  pressure 
tends  to  assume  a  circular  cross-section  A  cylinder  of  nominal 
circular  section,  but  departing  from  the  true  form  to  some  extent, 
tends  to  correct  this  departure  under  internal  pressure  ;  or  if  a 


—  68 

circular  cylinder  under  internal  pressure  is  deformed  by  any  ex- 
ternal force,  it  tends  to  resume  its  circular  shape.  Thus  a  circular 
cylinder  under  internal  pressure  is  in  "  stable  equilibrium."  If 
the  section  is  other  than  a  true  circle  there  is  a  flexure  stress,  as 
well  as  tension,  when  under  pressure. 

29.  Resistance  of  Thick  Cylinders  under  Internal  Press- 
ure.— [Unwin,  §  26  a,  page  48].  If  the  walls  of  the  cylinder  are 
quite  thick  relative  to  its  diameter,  the  intensity  of  stress  is  not 
uniform  across  any  section.  The  inner  layers  are  strained  most, 
because  it  is  by  yielding  of  the  inner  layers  that  stress  is  induced 
in  the  outer  layers  The  Grashof  formulas,  Unwin,  page  48, 
equations  (3)  and  (33),  may  be  used  for  thick  cylinders. 

Kquation  (3)  is  to  be  used  in  the  general  case.  For  con- 
venience it  is  given  here. 

\  (I) 

J 

Example  : — The  cast  iron  cylinder  of  a  hydraulic  press  is  8" 
diameter;  intensity  of  fluid  pressure  is  to  be  1000  Ibs.  per  sq. 
inch  ;  safe  tensile  stress  in  material  taken  at  2,500  Ibs.  per  sq. 
inch.  Required  the  thickness  of  walls.  By  eq.  (3), 


j-i-i-    /3X  ^2.6  inches. 

2     (.  ^3X2,500  —  4X1,000) 


The  thin  cylinder  formula, — eq.  (3),  art.  27,— if  applied  to  this 
problem,  would  give  a  thickness  of  only  1.6  inches. 

An  examination  of  the  above  formula  shows  that  if  4p  =  3f 
(^p  =.  y^f)  the  thickness  would  be  infinite.  This  does  not  mean 
that  a  pressure  of  1875  pounds  per  square  inch  would  necessarily 
burst  the  cylinder,  in  the  preceding  example,  but  it  indicates  that 
this  pressure  would  produce  a  stress  exceeding  2.500  Ibs.  maxi- 
mum intensity,  in  a  cylinder  of  8  inches  diameter,  however  thick 
it  may  be. 

30.  Resistance  of  Thin  Cylinders  to  External  Pressure. 
— [Unwin,  §40-41,  page  82-85.]  A  similar  analysis  to  that  given 
for  cylinders  under  internal  pressure  (art.  27,)  could  be  applied  to 


-69- 

this  case.  As  the  normal  pressures  are  reversed  in  direction  the 
stresses  produced  are  of  opposite  sign  to  those  of  the  former  case  ; 
or  the  stress  in  the  walls  of  the  cylinder  becomes  compression 
under  the  application  of  external  pressure. 

If  the  non-circular  cylinders  of  either  Figs.  28  or  31  be  con- 
sidered as  subjected  to  external  pressure,  the  force  tending  to  in- 
crease the  major  axis  will  be  seen  to  be  greater  than  those  tend- 
ing to  increase  any  shorter  axis  ;  hence,  the  external  pressure  will 
cause  collapse  of  the  cylinder,  unless  the  flexural  rigidity  of  the 
material  is  sufficient  to  prevent  this  action.  In  a  cylinder  of 
nominal  circular  section,  any  departure  from  the  ideal  section 
would  be  increased  by  the  external  pressure  :  Or,  if  a  cylinder  of 
true  circular  section  is  deformed  in  any  way  while  under  external 
pressure,  this  pressure  would  tend  to  still  further  increase  the  de- 
formation. In  other  words,  the  cylinder  under  external  pressure 
is  in  "  unstable  equilibrium."  As  perfectly  true  circular  sections 
and  homogeneous  materials  are  not  attainable  under  the  condi- 
tions of  practice,  the  danger  of  collapse  must  often  be  taken  into 
account  in  designing  a  pipe,  tube  or  flue,  to  withstand  external 
fluid  pressure.  The  resistance  of  the  heads  or  the  flanges  at  the 
ends  reinforces  the  shell  against  collapse  more  in  a  short  cylinder 
than  in  a  longer  one  of  similar  diameter.  For  this  reason,  the 
resistance  to  collapse  is  a  function  of  the  length  as  well  as  of  the 
diameter  and  thickness  of  walls.  Unwin  shows  (page  83)  the 
characteristic  forms  of  collapsed  fluts  for  given  ratios  of  length 
t<>  diameter  ;  and  he  has  deduced  a  theoretical  formula,  eq.  (23), 
based  upon  Euler's  formula  for  long  columns.  The  condition  of 
a  circumferential  strip  of  the  cylinder  under  external  pressure  is 
similar  to  that  of  a  long  column,  and  just  as  Euler's  formula  ordi- 
narily gives  too  high  a  value  for  the  strength  of  a  column,  eq. 
(23)  of  Unwin  is  found  not  to  accord  very  closely  with  observed 
results  on  the  resistance  to  collapse  of  flues 

31.  Collapse  of  Boiler  Flues  ;  Collapse  Rings  — [Unwin, 
§§  41-42,  pages  84-88.]  Fairbairn  derived  an  empirical  formula 
from  his  experiments  on  collapsing  tests  of  flues,  which  is  given 
by  Unwin,  eq.  (24),  page  84.  This  indicates  that  the  collapsing 


-70- 

pressure  does  not  increase  as  rapidly  as  the  cube,  but  only  a  litt!e 
more  rapidly  than  the  square  of  the  thickness.  The  more  con- 
venient approximate  expression 


is  frequently  used  in  connection  with  computations  of  boiler  flues. 
Fairbairn's  value  of  the  coefficient  C,  is  9,672,000  for  collapsing 
pressure,  and  Unwin  gives  it  as  3,500,000  for  working  pressure, 
[see  eq.  (b  ,  page  88].  This  last  value  is  obtained  from  examina- 
tion of  actual  flues  "30  feet  length  and  30  to  36  inches  in 
diameter,"  probably  of  the  Lancashire  type  of  boiler  ;  but  the 
factor  of  safety  (rather  less  than  3),  is  certainly  low.  The  Lloyds 
regulation  (British)  allows  the  following  pressure  in  boiler  flues 
and  furnaces 


(a) 

1037 

all  dimensions  in  inches,  and  pressure  in  pounds  per  square  inch. 
The  U.  S.  Board  of  Supervising  Inspectors  of  Steam  Vessels 
(U.  S.  B.  S.  I.)  has  adopted  these  same  regulations. 

The  Rules  also  provide  that,  in  flues  reinforced  by  collapse 
rings  of  specified  dimensions,  the  length  between  such  rings  is  to 
be  taken  as  the  length  of  the  flue  in  the  above  formulas.  [Con- 
sult Rules  U.  S.  B.  S.  I.  of  Steam  Vessels,  §433;  also  Unwin 
§  42,  pages  85-86]. 

If  a  cylinder  under  external  pressure  could  be  depended  upon 
to  fail  only  by  actual  crushing,  instead  of  through  collapse 
(buckling)  the  formula 

>  =  '?  (3) 

would  apply,  as  in  internal  pressure  [seeeq.  (2),  art.  27]  ;  remem- 
bering that  the  stress  (_/)  is  compression  under  external  pressure. 
If  eq.  (3)  gives  a  lower  working  pressure  than  eq.  (i),  the  flue 


designed  by  eq.  (3)  will  be  safe  against  collapse,  (see  Unwin, 
page  88).  If  /"be  taken  at  4,000  (a  value  allowed  by  the  British 
Board  of  Trade  and  rather  less  than  that  allowed  for  lap  welded 
or  riveted  flues  by  the  U.  S.  B.  S.  I.)  equation  (3)  may  be  used 
when  /  <  134.4,  for  when 

_  2  ft 8,000  /  ^  1,075,200 12 

p  ~ ''  ~T  ~~  ~d~        ~Jd~ 

.-.  /<  134.4'- 

32.  Corrugated  Furnace  Flues. — Flues  corrugated,  as  in 
Fig.  33,  are  very  much  stiffer  against  collapse  than  plain  cylin- 
drical flues,  and  may  be  safely  made  of  any  desired  length,  with 
proper  dimensions  of  corrugations.  When  the  corrugations  are 
not  less  than  i]4  inches  deep,  not  more  than  8  inches  centre  to 
centre  of  corrugations,  and  plain  portions  at  the  ends  do  not 
exceed  9  inches,  the  U.  S.  B.  S.  I.  allows  a  working  pressure  of 


This  is  also  the  formula  used  by  the  British  Board  of  Trade. 


V. 
RIVETED    JOINTS. 


33.  General   Considerations   of   Riveted  Joints  — [Unwin, 
§§  47-48.  pages  95-102.] 

34.  Size  of  Rivets  for  Plates  of  different  Thicknesses.— 
[Unwin,  §  49,  pages  102-103.] 

In  punching  plates  which  are  quite  thin,  relative  to  the  diameter 
of  the  punch,  the  action  approaches  pure  shearing,  and  the  rela- 
tion given  by  Unwin  on  page  102  is  correspondingly  exact.  If, 
however,  the  thickness  is  so  great  that  the  pressure  between  the 
end  of  the  punch  and  the  plate  reaches  the  intensity  at  which  the 
metal  of  the  plate  will  flow,  the  hole  may  be  formed  by  a  com- 
bined lateral  flow  and  a  shear.  Time  is  required  for  the  change 
iu  molecular  arrangement  which  occurs  during  the  flow  of  a 
ductile  metal  ;  but  if  the  motion  of  the  punch  is  not  too  rapid, 
good,  ductile  wrought  iron,  or  soft  steel,  will  flow  under  the 
punch,  before  the  crushing  stress  of  a  properly  tempered  tool  is 
reached.  It  is  thus  possible  to  force  a  punch  through  a  plate  of 
such  material  when  the  thickness  of  plate  is  several  times  the 
diameter  of  the  punch.  Hoopes  &  Townsend,  of  Philadelphia, 
punched  holes  y7^"  diameter  in  wrought  iron  i^"  thick,  and  it  is 
stated  that  a  single  punch  made  585  of  such  holes.  In  this  case 
the  pressure  on  the  end  of  the  punch  would  have  been  about 
650,000  Ibs.  per  square  inch,  had  the  metal  been  simply  sheared  ; 
while  it  is  probable  that  flow  began  under  a  pressure  of  about 
one-tenth  this  intensity. 

The  lateral  flow  of  the  metal  in  this  instance  was  evidenced  by 
the  fact  that  the  "wad,"  or  punching,  from  one  of  these  holes 
did  not  contain  one-half  as  much  metal  as  was  displaced  in  mak- 
ing the  hole. 

The  pressure  of  flow  of  ductile  wrought  iron  and  mild  steel  is 


-73  — 

probably  not  ordinarily  over  60,000  to  70,000  Ibs.  per  square  inch, 
which  is  well  within  the  crushing  resistance  for  tempered  tool 
steel,  in  the  absence  ot  severe  shock. 

The  injury  to  plates  by  punching,  to  which  Urnvin  refers  on 
page  96,  is  due  to  this  lateral  flow  ;  and  it  becomes  more  import- 
ant with  an  increase  in  thickness  of  the  plates. 

35.  Lap   of  Plates  and   Pitch  of   Rivets. — [Unwin,   §   50, 
page  103.] 

36.  Forms   of   Riveted   Joints. — [Unwin,    §  51,   pages    104- 
107.]     The  figures  in  this  section  show  several  standard  forms  of 
riveted    joints.     Numerous   other   forms   are    in    use ;    but    the 
methods  of  computation  to  be  given  in  art.   38  may  readily  be 
extended  to  cover  any  case. 

37.  Modes  of  fracture  of  Riveted  Joints.— [Unwin,  §§  52- 
53,  pages  107-108  ]     Of  the  four  methods  of  fracture  mentioned 
by   Unwin,  the  third  (breaking  out  of  the  plate  in  front  of  the 
riret)  may  always  be  avoided  by  giving  sufficient   lap.      Hence 
eq.  (4)  of  §  r,3  (Unwin)  may  be  neglected. 

The  only  objection,  in  practice,  to  large  lap  is  the  somewhat 
greater  difficulty  in  securing  a  tight  joint  by  caulking. 

The  resistance  to  the  tearing  of  the  plates,  shearing  of  rivets, 
or  crushing  of  the  rivets  (or  plates)  are  inter-dependt-nt,  and  for 
given  materials  of  plates  and  rivets  there  are  definite  relations 
between  the  pitch  and  diameters  of  rivets,  for  any  given  form  of 
joint,  which  cannot  be  departed  from  without  sacrifice  of  strength 
in  the  joint  as  a  whole.  This  will  appear  from  the  discussion  in 
the  next  article. 

38.  Strength  of  Riveted  Joints  — Any  riveted  joint  may  be 
considered  as  consisting  of  a  number  of  unit  strips,  all  alike  as  to 
width,   number  of  rivets,  pitch,  etc.     Thus,   in  a  single  riveted 
joint  each  strip  has  a  width  equal  to  the  pitch  of  the  rivets,  and 
contains  one  rivet,  (see  Unwin,  Figs.  48  to  51,  page  107.)     With 
the  ordinary  double    riveted  joint,   the   unit  strip  contains  two 
rivets  and  its  width  is  equal  to  the  pitch  of  rivets  in  either  row. 
Figs   44  and  45  (Unwin)  show  portions  of  double  riveted  joints 
each  roughly  equal  to  two  such  unit  strips.     A  similar  division  of 


-74- 

the  joint  into  unit  strips,  whatever  its  form,  is  evidently  possible. 
The  computations  for  strength  against  tearing  of  the  plates,  shear- 
ing of  rivets,  etc  ,  may  be  made  for  such  a  unit  strip,  as  every 
other  equal  strip  would  have  the  same  computed  strength. 

The  following  notation  will  be  used  throughout  this  article  : 

d=  diameter  ot  hole,  (diam.  of  rivet  when  upset.) 

p=  pitch  of  rivets  along  a  row. 

/=  thickness  of  the  plates. 
ft=  tensile  strength  of  the  plates  per  sq.  in. 
f0  =  crushing  strength  of  the  rivets,  or  plates,  per  sq.  in. 
fa  =  shearing  strength  of  rivets  in  single  shear  per  sq.  in.* 

fs'  =  shearing  strength  of  rivets  in  double  shear  per  sq.  in. 

C==  Crushing  resistance  of  rivets,  or  plates,  per  unit  strip. 

S=  Shearing  resistance  of  rivets  per  unit  strip. 

T=  Tensile  resistance  of  plates  (net  section)  per  unit  strip. 

P=  Tensile  resistance  of  plates  (solid  section)  per  unit  strip. 

E=  Efficiency  of  joint. 

The  following  clearly  indicates  the  meaning  of  these  symbols  : 

I.    SINGLE  RIVETED  LAP  JOINTS. 

[See  Umvin,  Figs.  48  to  51  ;  also  equations  (2),   (3)  and  (5), 
§53-] 

The  net  tensile  strength  of  the  strip  (through  the  rivet)  is 

r=  (>-*)//,.  (i) 

The  shearing  strength  of  the  rivet  is 

S=  -<*'./.=  -7854  *%  (2) 

The  crushing  strength  of  rivet,   or  plate,   whichever  has  the 
lower  crushing  resistance,  is 

C=dtfc.  (3) 


*  Experiments  show  that  rivets  in  double  shear  do  not  usually  have  as 
great  strength  (per  square  inch  sheared)  as  similar  rivets  in  single  shear. 
Or  the  resistance  of  the  two  sections  in  double  shear  is  not  twice  that  of  the 
one  section  in  single  shear.  On  the  other  hand,  the  crushing  resistance  is 
probably  higher  in  double  shear  rivets,  because  of  their  more  uniform  bear- 
ing. This  is  neglected  in  the  present  article. 


-75  — 

The  tensile  strength  of  the  solid  plate  ;  that  is,  of  a  section 
across  the  strip  not  passing  through  a  rivet  hole,  is,  per  unit  strip 

P  =  fitfi.  (4) 

The  efficiency  of  the  joint,  £,  is  the  ratio  of  the  strength  of  the 
joint  to  the  strength  of  the  solid  plate  ;  or  it  is  the  smallest  of  T, 
S,  or  C,  divided  by  P.  For  highest  efficiency,  T,  S  and  C  should 
be  equal.  If  the  proportions  are  such  that  this  result  is  attained, 
the  three  equations  (i),  (2)  and  (3)  involve  the  three  unknown 
quantities  :  /»,  d,  and  T-~S=  C;  the  terms  /./»./,  and/e  being 
known.  Equating  ,5"  and  C,  eqs.  (2)  and  (3). 

'«/'/.=  </£.       ••  rf=i.27/-/«  (5) 

4  A 

This  expression  gives  the  proper  theoretical  diameter  of  rivets 
for  plates  of  a  given  thickness,  when  the  values  of  fa  and  fc  are 
fixed.  Equating  7"and  S,  eqs.  (i)  and  (2) 


This  gives  the  proper  theoretical  pitch. 

Equation  (5)  generally  gives  a  diameter  which  does  not  ex- 
actly agree  with  regular  "shop  dimensions,"  hence  the  diameter 
of  actual  rivets  (holes)  would  be  varied  to  give  a  convenient  size. 
Likewise,  the  pitch  would  perhaps  be  made  somewhat  different 
from  that  computed,  for  convenience  in  laying  off,  or  to  make 
even  spacing  in  the  entire  length  of  the  plate.  These  changes 
will  destroy  the  exact  equality  between  T,  S  and  C.  The  sub- 
stitution of  the  final  practical  values  of  d  and  p  in  equations  (i), 
(2),  (3)  and  (4)  will  give  the  values  of  T,  S,  C  and  P,  and  the 
smallest  of  the  first  three  of  these  values  divided  by  the  value  of 
P  gives  the  efficiency,  E. 

Example  :  A  single  riveted  lap  joint  is  to  be  designed  for  a 
plate  Y  inch  thick.  Tensile  strength  of  plate  —  58,000  pounds 
per  square  inch  ;  shearing  strength  of  rivets  (single  shear)  = 
40,000  ;  and  crushing  resistance  taken  at  70,000. 

d=  1.27  X  .5  X  70,000-^-40,000  =  1.  1  1  inches. 


-76- 

The  rivet  (hole)  will  be  taken  as  ly'j  inch  diameter  (i  inch 
rivets).  Using  this  value  of  d  in  eq.  (6) 

7854  X  289  X  40.000  +  17  =I22+I0628-^ 
.5  X  256  X  58,000          16 

The  pitch  may  be  taken  at  2fV,  unless  some  slightly  different 
pitch  would  space  to  better  advantage.* 
From  eq.  (  i  ) 

T=  (2.31  —  i.  06)  X  .5  X  58,000=  36,300  Ibs. 
From  eq.  (2) 

5  =  .7854  X  i  .  13  X  40,000  =  35,400  Ibs. 
From  eq.  (3) 

C=  1.0625  X  -5  X  70,000=  37,200  Ibs. 
From  eq.  (4) 

P=  2.31  X  .5  X  58,000  =  67,000  Ibs. 

.'.  The  efficiency  of  the  joint  equals  tbe  smallest  of  these  resist- 
ances, 5  in  this  case,  divided  by  the  strength  of  the  solid  plate,  P. 

.'.  E=S-^-  P  =  35,  400  -5-67.  000=  .53, 
or  the  efficiency  is  53  per  cent. 

II.    DOUBLK    RIVETED    LAP  JOINTS. 

The  unit  strip  contains  two  rivets,  one  in  each  row  ;  but  its 
cross-section  through  either  rivet  is  only  weakened  by  a  single 
hole,  hence 


The  shearing  resistance  of  the  two  rivets  in  this  unit  strip  is 

s-2z*ym=i.57*V.          .  w 

The  crushing  resistance  of  the  two  rivets  is 

C=2dtfc  (9) 

The  tensile  strength  of  the  solid  plate  per  unit  strip  is 

(10) 


*It  is  preferable  to  slightly  increase,  rather  than  to  decrease  the  pitch  as 
computed  ;  because  corrosion  weakens  the  plates  more  than  it  does  the 
rivets. 


-77  — 
Equating  6*  and  C,  eqs.  (8)  and  (9) 


.       . 

which  gives  the  same  diameter  of  rivet,  for  the  same  thickness  of 
plates  and  materials,  as  in  the  case  of  a  single  riveted  lap  joint. 
Equating  T  and  6",  eqs.  (7)  and  (8), 

(p-dWl-l'Slff.'-'P^^&f^+d  (12) 

The  process  in  designing  this  joint  is  similar  to  that  given  for 
single  riveted  joints.  First,  the  diameter  of  rivet  is  computed 
from  eq.  (n),  and  some  regular  size  of  nearly  this  diameter  is 
adopted.  Second,  the  pitch  is  computed  from  eq.  (12),  and  this 
may  be  modified  to  give  a  convenient  dimension  Then  7",  S, 
and  Care  computed  from  eqs.  (7),  (8)  and  (9),  respectively,  and 
the  smallest  of  these  divided  by  P,  —  as  found  from  eq.  (10).  — 
gives  the  efficiency. 

III.    TRIPLE  RIVETED  LAP  JOINTS. 

If  a  triple  riveted  lap  joint  is  made  with  three  rows  of  rivets  (all 
rows  having  the  same  pitch),  there  would  be  three  rivets  in  each 
unit  strip  of  width  equal  to  the  common  pitch.  The  equations 
would  then  be 

T=(p-d}tfl  (13) 


4 

(15) 

/>=///,  06) 

d=i.27fS°  (17) 


p  = 


2,356  rfy,  +  d  (I8) 

V» 

And  the  design  would  be  carried  out  as  in  the  preceding  cases. 
It  will  be  shown,  however,  that  a  higher  efficiency  is  obtained 


-78- 

by  giving  the  joint  the  form  shown  in  Fig.  46  (Unwin),  in  which 
the  two  outer  rows  have  twice  the  pitch  of  the  inner  rows.  In 
this  second  form  of  the  triple  riveted  joint,  there  are  four  rivets 
per  unit  strip  ;  the  width  of  this  strip  being  p'  —  2p,  in  which  p 
is  the  pitch  of  the  middle  row.  For  such  a  unit  strip 


(14) 

(15') 
06') 
Equating  S'  and  C'}  eqs.  (14')  and  (15') 

(17') 


..  .. 

Equating  T'  and  S',  eqs.  (13')  and  (14') 


These  equations  are  used  as  in  the  preceding  cases.  First, 
finding  d  from  eq.  (17')  ;  then  finding  p'  from  eq.  (18')  ;  finally 
computing  7"',  S',  C'  and  P'  ,  and  getting  the  efficiency  by  divid- 
ing the  smallest  of  the  first  three  resistances  by  P'  . 

To  show  that  this  modified  triple  riveted  joint  (Fig.  46,  Unwin) 
is  more  efficient  than  three  rows  of  rivets  with  equal  pitch,  the 
general  expressions  for  efficiency  of  each  form  will  be  derived.  It 
will  be  assumed  that  each  joint  is  of  maximum  strength  for  its 
form;  that  is.  that  T---=S=.C,  and  that  T  =  S'=C.  The 
efficiency  would  thus  be  given  for  the  first  form  by  dividing  7\  S, 
or  C  by  P  ;  and  for  the  second  case  by  dividing  T'  ,  S',  or  C' 
by  P'. 

For  the  first  form  of  joint,  from  eqs.  (14),  (16)  and  (18) 


(a) 


—  79  — 
For  the  second  form  of  joint,  from  eqs.  (14'),  (16')  and  (18') 


Since  tlie  only  difference  in  the  two  equations  (a)  and  (a')  is 
that  the  latter  has  the  smaller  denominator,  E'  is  greater  than  E; 
or  the  joint  with  outer  rows  having  twice  the  pitch  of  the  inner 
rows  is  of  higher  efficiency  than  the  form  with  the  same  pitch  in 
all  rows,  provided  each  joint  is  designed  for  its  maximum 
efficiency. 

Butt  joints  with  a  single  covering  strip,  or  welt,  (Unwin,  Figs. 
43  and  45)  are  sometimes  used  for  circumferential  seams  of  a 
boiler  shell  ;  or,  (with  countersunk  rivets,  Unwin,  Fig.  39),  in 
ship  work,  where  a  smooth  exterior  surface  is  desired.  Single 
welt  butt  joints  would  be  designed  by  the  formulas  for  lap  joints 
with  the  corresponding  number  of  rows  of  rivets  ;  for  the  cover- 
ing strip  forms  a  lap  joint  with  each  of  the  adjacent  plates. 

IV.    SINGLE  RIVETED  BUTT  JOINTS,  TWO  WELTS. 

This  joint  is  not  very  commonly  used,  unless  in  the  cir- 
cumferential seams  of  boiler  shells  which  have  double  riveted 
butt  joints  with  two  welts  for  the  longitudinal  seams.  As 
the  circumferential  seams  require  only  half  the  strength  of 
the  longitudinal  seams  (see  art.  27),  it  is  the  common  prac- 
tice to  make  the  circumferential  joints  of  a  form  having 
lower  efficiency  than  the  longitudinal  joints.  Even  single 
riveted  lap  joints  have  an  efficiency  greater  than  50  per  cent. 
(with  any  ordinary  materials)  ;  hence  they  might  be  safely  used 
for  circumferential  seams  with  any  possible  efficiency  in  the 
longitudinal  joints.  Convenience  often  dictates  a  butt  joint  for 
circumferential  seams,  as  this  reduces  the  difficulty  of  disposing 
of  the  welts  of  the  other  joints,  (consult  Unwin,  Fig.  58.) 

In  joints  with  two  welts,  each  somewhat  thicker  than  one  half 
the  main  plates,  the  bearing  area  of  the  rivets  against  the  plates 


(which  determines  the  resistance  to  crushing)  is  dt,  as  in  lap 
joints  ;  but  each  rivet  presents  two  ctoss-sections  to  be  sheared. 
As  stated  in  the  footnote  on  page  74,  the  resistance  of  a  rivet  to 
double  shear  cannot  be  assumed  at  twice  the  resistance,  per  unit 
of  area,  of  the  same  rivet  in  single  shear.  The  notation  at  the 
head  of  this  article  gives  fs'  as  the  symbol  for  unit  shearing  stress 
in  double  shear. 

The  single  riveted  butt  joint  with  two  welts  is  similar  in  form 
to  that  shown  by  Unwin,  Fig.  47,  except  that  there  is  only  one 
row  of  rivets  each  side  of  the  butt,  and  the  welt  is  correspond- 
ingly narrower.  In  such  a  single  riveted  butt  joint,  the  unit 
strip  has  a  width  />,  and  it  contains  one  rivet  (each  side  of  the 
butt)  which  is  in  double  shear.  The  equations  are  as  follows  : 

(19) 

(20) 

(21) 

(22) 

Equating  5  and  C,  eqs.  (20)  and  (21) 

(23) 


which  gives  the  proper  diameter  of  rivets,  to  be  modified,  as  be- 
fore, for  convenient  shop  dimensions. 
Equating  7"  and  5,  eqs.  (19)  and  (20) 

(p-d)tf,=  *«*'/;    .-.  p=l-V£fL  +  d          (24) 

2  *J\ 

Taking  the  nearest  convenient  pitch  to  that  computed  by  eq. 
(24),  the  values  of  T,  S,  C  and  P  are  found  as  in  the  preceding 
cases,  and  the  smallest  of  the  first  three  divided  by  P  gives  the 
efficiency. 

V.    DOUBLE  RIVETED  BUTT  JOINTS,  TWO  WELTS. 

In  this  joint,  if  the  rivets  in  each  row  have  the  same  pitch,  the 
unit  strip  has  a  width  equal  to  the  pitch  ;  each  unit  strip  con- 
tains two  rivets  in  double  shear,  and  the  equations  are 


tft  (25) 

*f;=ird*f;  (26) 

(27) 
(28) 

Equating  .S  and  T,  eqs.  (26)  and  (27) 


Equating  .S"  and  7",  eqs.  (25)  and  (26) 

p="d^+d  (30) 

If  the  double  riveted  butt  joint  be  made  like  that  shown  by 
Fig.  47  (Unwin)  with  the  outer  rows  twice  the  pitch  of  the  inner 
rows,  the  efficiency  of  the  joint  may  be  increased,  as  in  the  simi- 
larly modified  triple  riveted  lap  joint.  This  modified  double 
riveted  butt  joint  has  a  unit  strip  of  width  p  '  =  2/»,  p  being  the 
pitch  of  the  rivets  in  the  inner  rows.  Each  unit  strip  contains 
three  rivets  in  double  shear.  The  equations  for  this  joint  are 

r  -(/-*)//,  (25') 


(27') 
P'  =  P'  t/t  (28') 

Equating  S'  and  C',  eqs.  (26')  and  (27') 

d—.e^GtS*  (29') 

Equating  T'  and  5',  eqs.  (25')  and  (26') 

(30') 


—  82  — 
VI.    TRIPLE   RIVETED   BUTT  JOINTS,    TWO   WELTS. 

The  joint  to  be  considered  is  the  modified  form  shown  by  Fig. 
34.  It  consists  of  two  rows  of  rivets  in  double  shear  and  one 
outer  row  in  single  shear,  each  side  of  the  butt,  the  pitch  of  rivets 
in  the  outer  rows  being  twice  that  of  the  inner  rows.  One  of  the 
covering  strips  is  only  wide  enough  to  take  the  two  close  pitch 
rows,  while  the  other  strip  is  wide  enough  for  the  three  rows  each 
side  of  the  butt.  This  form  of  joint  has  a  high  efficiency,  and  is 
much  in  favor  for  the  best  class  of  work  under  heavy  pressure. 
The  unit  strip  contains  four  rivets  in  double  shear  and  one  in  sin- 
gle shear,  and  its  width  equals  the  pitch  of  outer  row,  p'  '  =  2  p. 

The  equations  are 

(31) 

(32) 

(33) 

P  =  P'tf,-  (34) 

Equating  6"  and  C,  eqs.  (32)  and  (33) 

1-59  */.    ->36</c 
/,'+  */.      8/'  +  /. 

Equating  T  and  .S,  eqs.  (31)  and  (32) 


_ 

There  are  many  other  possible  arrangements  of  rivets  in  a  joint, 
but  the  preceding  include  most  of  the  usual  forms,  and  the  deriva- 
tion of  the  above  equations  will  suggest  the  method  of  finding 
those  for  any  given  style  of  joint. 

39.  General  Equations  for  Riveted  Joints.—  The  funda- 
mental equations  for  efficiency  of  riveted  joints  of  various  styles, 
as  well  as  those  for  the  diameter  and  pitch  of  the  rivets,  may  be 


-83- 

put  in  more  general  forms  than  those  of  the  preceding  article. 
The  equations  developed  in  the  present  article  are  applicable  to 
any  style  of  riveted  joint. 

The  unit  strip  is  of  width  equal  to  the  pitch  ;  the  maximum 
pitch  being  taken  for  such  width  of  unit  strip  if  all  rows  do  not 
have  the  same  pitch,  as  in  the  modified  double  and  triple  riveted 
joints. 

The  general  expression  for  the  net  tensile  strength  of  the  unit 
strip  is 

T=(p  —  dVfv  (i) 

The  general  expression  for  resistance  to  shearing  of  the  rivets 
in  the  unit  strip  is 

s=ui*.1/.  +*—--/:  co 

4  4 

in  which  n  equals  the  number  of  rivets  in  single  shear  and  m 
equals  the  number  of  rivets  in  double  shear. 

The  general  expression  for  resistance  to  crushing  of  the  unit 
strip  is 

C=ndtfc  +  mdtK.  (3) 

This  is  a  recognition  of  the  fact  that  the  resistance  to  crushing 
with  single  shear  (/,.)  is  probably  less  than  the  resistance  with 
double  shear  (_/j')>  owing  to  the  more  uniform  distribution  of 
bearing  pressure  in  the  latter  case  ;  such  a  distinction  was  not 
made  in  the  equations  of  the  preceding  article. 

The  tensile  resistance  of  the  solid  strip  is 

P  =  P*fv  (4) 

Equating  S  and  C,  eqs.  (2)  and  (3) 


4 

.     ,         ir<t*(nfs+  2m  //) 

0/0    +    «/.') 

Equating  Tand  5,  eqs.  (i)  and  (2)  and  solving 

~-  (»/.  +  2  »*/.')  +  <///-  P.  (6) 

4 


Q. 
04 

If  the  joint  is  designed  for  maximum  efficiency,  T=  S=  C, 
hence  any  one  of  these  three  quantities  divided  by  /'gives  the 
efficiency  of  the  ideal  joint,  for  any  given  form,  or  dividing  (2) 
by  (6) 


4 
Substituting  the  value  of  dt  as  given  by  eq.  (5)  and  dividing 

numerator  and  denominator  by  -  —  (nfa  -f  2  mfs'), 

4 


+ 


(7) 


»/e  -f  mfe' 
If  all  the  rivets  are  in  single  shear 


/, 


If  all  the  rivets  are  in  double  shear 

i 


(7") 


Or  if  fe  =/c',  (as  assumed  in  art.  38),  and  the  number  of  rivets 
in  the  unit  strip,  n  +  m,  be  called  A", 

E=^r: 
w* 

This  holds  for  any  form  of  the  joint.  The  formula  (7),  or  the  appro- 
priate modified  form  of  it  (7'),  (7*),  or  (7""),  may  be  used  to  find 
the  maximum  possible  efficiency  of  any  form  of  riveted  joint 
when  the  resistances  to  tension  and  crushing  are  known.  It  will 
be  noticed  that  the  resistance  of  the  rivets  to  shearing  does  not 
appear  in  this  general  formula  for  maximum  efficiency.  It  is  thus 


-85- 

seen  that,  with  given  resistance  to  tension  and  crushing,  the 
shearing  strength  of  the  rivets  does  not  affect  the  attainable 
efficiency.  However,  the  shearing  resistance  of  the  rivets  does 
affect  the  proportions  of  the  joint^necessary  for  such  maximum 
efficiency. 

This  formula  is  useful  in  finding  the  limiting  efficiency  of  joint 
for  any  form  and  materials ;  the  actual  proportions  adopted  may 
give  a  lower  efficiency,  but  can  never  give  higher  efficiency. 

It  is  possible  to  derive  a  set  of  general  expressions  for  the 
diameter  of  rivets  and  for  their  pitch  which  can  be  used  for  any 
form  of  joint.  The  notation  used  is  the  same  as  in  the  preceding 
work  of  this  article. 

Equating  S  and  6",  eqs.  (2)  and  (3)  and  solving  for  d 

d^^x^+^Kt  (8) 

TT       nfa  +  2  mf, 

which  gives  the  proper  diameter  of  rivets  for  a  given  thickness  of 
plate,  when  the  number  of  rivets  in  single  shear  and  the  number 
in  double  shear  and  the  corresponding  shearing  and  crushing 
resistance  are  known. 

Equating  T and  C,  eqs.  (i)  and  (3),  and  solving  for/ 


(9) 
Or,  equating  T  and  S,  eqs.  (i)  and  (2),  and  solving 


In  case  of  simple  lap  joints,  all  rivets  are  in  single  shear,  hence 
d=ft  (8') 


p  =  »*d+d  (9') 

Or   p  =  "d*nf*  +  d  (10') 


—  86  — 

In  case  of  simple  butt  joints  with  two  welts  through  which  all 
the  rivets  pass,  the  livets  are  all  in  double  shear,  hence 

<*=-•£/  (8") 


Or   p  =  '"^nLL^  d*  +  d  (10") 

If,  in  the  general  form  of  joint,//  is  taken  as  equal  to/,  and 

f,'-/. 

d=^%+^!  (8"> 


Or   /  =  *  d*  °  +  rf  (io'") 

4  Vt 

These  general  equations  are  due  to  Mr.  William  N.  Barnard, 
who  also  suggested  the  above  expressions  for  the  maximum 
efficiency  in  the  general  case. 

40.  Customary  Proportions  of  Riveted  Joints.  —  The  method 
of  computation  indicated  in  Article  38  should  be  used  when 
the  necessary  data  is  available,  especially  for  joints  subjected 
to  high  pressure.  It  is  apparent  that  any  variation  in  the 
strength  of  the  material  used  would  affect  the  proportions  when 
such  methods  are  employed  in  designing  a  joint  for  maximum 
efficiency,  and  that  a  table  of  rivet  diameters  and  pitches  would 
have  to  be  very  extensive  to  cover  the  entire  range  of  practice. 
However,  it  is  not  uncommon  to  adopt  regular  diameters  and 
pitches  for  a  given  thickness  of  plate  and  form  of  joint  ;  of  course 
such  proportions  would  only  give  the  best  efficiency  for  certain 
combinations  of  shearing,  crushing  and  tensile  strengths.  The 
following  formulas  are  derived  from  Tables  given  by  the  Hartford 
Steam  Boiler  Inspection  and  Insurance  Co.  for  lap  joints,  using 
the  following  notation  : 


—  87  — 

d=  diameter  of  hole  =  diam.  of  rivet  +  y^  inch. 
p  =  pitch  of  rivets. 
/  =  thickness  of  plates. 
All  dimensions  in  inches.     Iron  rivets. 

For  Iron  Plates,  d=  t  +  J^".  (i) 

For  Steel  Plates,  d=t+  %" .  (2) 

Single  Riveted  Lap  Joints  p=t+  i^".  (3) 

Double  Riveted  Lap  Joints.  p  =  2t+2#".  (4) 

Various  hand-books  and  treatises  on  boiler  construction  give 
tables  of  this  character  ;  but  it  is  well  to  check  such  values  be- 
fore adopting  them  for  other  than  light  service. 

41.  Strength  of  Iron  and  Steel  used  for  Boiler  Plates  and 
Rivets.      [Unwin,  §  54,  page  109.]      Ductility  is  of  even  greater 
importance  than  strength  in  boiler  materials,   as   the   straining 
actions  due  to  pressure,  unequal  expansion,  etc.,  are  often  dis- 
tributed very  unequally.      Hence  it  is  important  to  use  materials 
which  will  distribute  the  stress,  through  yielding,  before  local 
rupture  occurs.     For  these  reasons  only  soft  wrought  iron  and 
steel  are  used  in  this  class  of  work.      The  tensile  strength  of  the 
usual  grades  of  boiler  plates  may  be  assumed  at  about  the  values 
given  by  Unwin  in  §  54. 

The  shearing  strength  of  wrought  iron  rivets  has  been  given 
by  Mr.  J.  M.  Allen,  President  of  the  Hartford  Steam  Boiler 
Insurance  and  Inspection  Co.,  at  38.000  Ibs.  per  square  inch  for 
single  shear,  and  at  35,000  Ibs.  per  square  inch  for  double  shear. 
The  values  are  based  on  tests  made  at  the  Watertovvn  Arsenal. 
Other  authorities  assume  somewhat  higher  values.  Wrought 
iron  rivets  have  remained  in  great  favor  even  with  the  general 
adoption  of  steel  plates  ;  but  steel  rivets  are  now  being  largely 
used.  Steel  rivets  may  be  assumed  to  have  a  strength  of 
45.000  Ibs.  per  square  inch  in  single  shear,  and  perhaps 
40.000  to  42,000  Ibs.  per  square  inch  in  double  shear.  The 
crushing  resistance  of  rivets  and  plates  is  not  quite  so 
defiinitely  known,  but  it  may  be  taken  at  from  60,000  to 
70,000  Ibs.  per  square  inch. 

42.  Apparent  Strength  of  Perforated   Plates.     [Unwin,  §§ 


54~55  J  pages  109-112.]  As  shown  by  Unwin  (in  the  discussion 
of  ' '  Tenacity  of  drilled  plates ' ' ,  and  ' '  Tenacity  of  punched 
filates"),  the  drilled  plates  generally  exhibit  a  higher  apparent 
unit  strength  than  ordinary  test  bars  of  the  same  material.  This 
is  because  the  drilled  plate  is  equivalent  to  a  row  of  short  speci- 
mens placed  side  by  side,  and  short  test  bars  usually  show  an 
excess  over  the  true  strength  of  the  material,  especially  with 
ductile  metals.  A  punched  plate  has  this  same  advantage  of 
form  ;  but  the  injury  to  the  plate  by  lateral  flow  in  punching 
often  more  than  compensates  for  the  gain  in  apparent  strength  due 
to  the  test  of  "short  specimens".  If  the  punched  plate  is 
annealed  or  reamed  out  after  punching,  it  may  show  an  increase 
in  apparent  strength  corresponding  approximately  to  that  observed 
in  drilled  plates.  In  the  computations  of  riveted  joints,  it  is  usual 
to  neglect  this  element  of  additional  strength. 

43.  Friction  and  Binding  of  Riveted  Joints  —[Unwin,  §  57, 
pages    112-113].     The   friction    between   the   plates   due   to  the 
tension  in  the  rivets,  produced  by  their  contraction  in  cooling, 
tends  to  increase  the  apparent  shearing  strength  of  the  rivets. 
This  very  uncertain  element  cannot  safely  be  depended  upon,  be- 
cause a  slight  yielding  of  the  plates  and  rivets,  even  under  the 
ordinary  service  load,  may  soon  greatly  reduce  the  normal  pressure 
between  the  contact  surfaces  of  the  plates.     On  the  other  hand, 
the  bending  of  the  plates,  in  lap  or  single  welt  butt  joints,  may  be 
an  important  source  of  weakness.     See  Unwin,  Figs.  42  and  43. 
Repetition  of  this  bending  action  may   cause  a  plate  to  break 
through  the  solid  section,  near  the  edge  of  the  overlapping  plate  ; 
and  this  is  especially  apt  to  occur  if  the  plate  has  been  scored  by 
careless  use  of  a  caulking  tool  with  a  sharp  corner.     The  round 
nose  caulking  tool  is  safer  as  well  as  more  effective. 

44.  Distance  between  Rows  of  Rivets. — In  multiple  riveted 
joints  the  distance  between  the  rows  of  rivets  should  he  sufficient 
to  insure  a  greater  strength  against  tearing  of  the  plate  along  a 
zig-zag  section  than  straight  through  the  rivets  of  a  single  row. 
The  net  zig  zag  section  should  be  at  least  iy$  times  the  straight 
section  ;  when  the  diagonal  pitch  (distance  from  centre  of  a  rivet 


-89- 

in  one  row  to  the  centre  of  the  nearest  rivet  in  the  adjacent  row) 
is  given  by  the  following  expression.  If  pl  =  the  diagonal  pitch, 
P  the  straight  pitch  of  the  inner  rows,  and  d  the  diameters  of 

the  rivets, 

A  =  f^  +  **  (0 

Unwin  gives  as  the  minimum  diagonal  pitch,  twice  the  diameter 
of  the  rivets,  (§  50,  Fig.  41)  ;  while  the  proportions  shown  in  Fig. 
47  (Unwin)  allow  a  distance  of  2  d  between  centres  of  the  parallel 
rows  of  rivets.  In  triple  riveted  butt  joints  the  outer  rows  are 
often  placed  somewhat  farther  from  the  middle  rows  than  the 
latter  are  from  the  inside  rows,  especially  when  the  outer  rows 
have  only  half  as  many  rivets  (twice  the  pitch)  as  the  other  rows. 

45.  Graphic  Method  of  designing  Joints. — [Unwin,   §  64, 
pages  1 1 8-1 2 ij. 

46.  Junction  of  three  or  more  Plates. — [Unwin,  §  69,  pages 
124-126]. 

47.  Connection  of  Plates  not  in  one  Plane. — [Unwin,  §§  70- 
71,  pages  126-130]. 

48.  Position  of  Rivets  in  Bars.— [Unwin,  §  72,  pages  130- 

131]- 

49.  Cylindrical    Riveted    Structures. —  [Unwin,    §§    73-74, 
pages  131-135]. 

50.  Stayed   Flat  Surfaces. — [Unwin,   §   75,  pages   135-137  ; 
also  §  46,  pages  93,  94].     Copper  fire,  boxes  and  stays  are  not  now 
extensively  used  in  this  country,  having  been  almost  completely 
superseded  by  iron  and  steel. 

Short  stay  bolts  between  parallel  plates,  as  in  the  "water-leg  " 
of  locomotive  boilers,  are  liable  to  fracture  from  the  relative 
motion  of  the  connected  plates  due  to  unequal  expansion.  The 
fracture  occurs  at  the  bottom  of  the  thread  near  one  of  the  plates, 
usually  at  the  outside  plate.  The  drilled  end  stay  (Unwin,  Fig. 
79)  is  an  effective  means  of  calling  attention  to  a  broken  stay  bolt, 
and  the  practice  of  drilling  the  stays  is  now  quite  common  in  the 
better  class  of  work. 

The  relations  given  near  the  bottom  of  page  136  (Unwin),  are 
not  generally  applicable,  under  authorized  rules,  for  flat  stayed 
surfaces. 


Solving  eq.  (3^)  —  Unwin,  page  93  —  for  the  pressure, 

' 


In  which  p  =  working  pressure  of  steam  in  pounds  per  sq.  inch  ,* 
/"  =  allowable  stress  in  plate  in  pounds  per  sq.  inch  ;  t  =  thick- 
ness of  the  plate  in  inches,  and  a  =  the  pitch  (distance  center  to- 
center)  of  the  stay  bolts  in  inches. 

If  /"be  taken  at  6,000  Ibs.  per  square  inch, 

9X6,000;*^  (2) 

2          d  a 

The  Lloyds  rule  for  flat  stayed  surfaces  is 


in  which 

C=  go  for  iron  or  steel  plates  T7¥  inch  thick  or  less,  with  stays- 
screwed  into  the  plates  and  riveted  over  at  ends. 

C=  100  for  iron  or  steel  plates  over  yV  inch  thick,  with  screwed 
stays  riveted  over. 

C=  1  10  for  iron  or  steel  plates  y7^  inch  or  less,  with  screwed 
stays  and  nuts. 

C=  120  for  iron  plates  over  T7¥  inch  thick,  or  for  steel  plates 
over  yZg-  inch  and  less  than  y9^  inch  thick,  with  screwed  stays  and 
nuts. 

C=  135  for  steel  plates  y9^  inch  thick  or  over,  with  screwed 
stays  and  nuts. 

C  '  =  140  for  iron  plates  with  double  nuts  (nuts  inside  and  out- 
side of  plate). 

C=  150  for  iron  plates  with  double  nuts  or  stays,  and  washers 
on  the  outside  of  at  least  half  the  thickness  of  the  plates,  and  of 
diameter  not  less  than  one  third  the  pitch  of  stays. 


*The  form  (i6/f)2  in  eq.  (3)  is  convenient  for  computations,  for  it  equals 
the  square  of  the  number  of  sixteenths  of  an  inch  in  the  thickness  of  the 
plate. 


C=  160  for  iron  plates,  with  double  nuts  or  stays,  and  washers 
riveted  to  outside  of  plates,  washers  having  a  diameter  not  less 
than  two-fifths  the  pitch  of  stays  and  thickness  of  at  least  one- 
half  the  thickness  of  the  plates. 

C  =  175  for  iron  plates,  with  double  nuts  or  stays,  and  washers 
riveted  to  outside  of  plates,  washers  having  diameter  at  least  two- 
thirds  the  pitch  of- stays  and  thickness  equal  to  that  of  the  plates. 

For  steel  plates,  except  for  combustion  chambers  directly  ex- 
posed to  the  heated  gases,  Cmay  be  increased  from  140  to  175, 
from  150  to  185,  from  160  to  200,  and  from  175  to  220,  in  the 
above  cases. * 

It  will  be  noticed  that  the  Lloyds  formula,  eq.  (3),  is  of  the 
same  form  ;is  that  given  in  eqs.  (i)  and  (2).  The  constant  of 
eq.  (2")  is  nearly  equivalent  to  105  in  eq.  (3). 

Umvin  gives  the  rule  of  the  Board  of  Trade  (British)  with  the 
values  of  the  constants,  on  pages  93  and  94.  This  rule  is  pre- 
ferred by  some  authorities  ;  but  it  is  not  quite  so  simple  in  form 
as  the  Lloyds  rule  and  as  the  latter  is  unquestionably  safe,  it 
will  be  used  in  this  work. 

The  stay  bolt  itself  should  have  a  minimum  area  of  cross- 
section  (at  the  bottom  of  the  threads)  sufficient  to  carry  a  load  = 
pa2,  with  a  stress  not  over  6,000  to  7,000  for  wrought  iron,  and 
7,000  to  8,000  for  steel. 

51.  Diagonal  Stays.— [Unwin,  §  76,  pages  137-138]. 
Gusset  stays  (Unwin,   Fig.  81)  are  liable  to  unequal    tension 

across  the  thin  broad  section,  so  that  the  maximum  intensity  of 
stress  at  the  edge  may  be  excessive,  even  though  the  nominal 
(mean)  intensity  of  stress  is  quite  moderate.  The  diagonal  stay 
(Unwin,  Fig.  80)  is  to  be  preferred  to  the  gusset  stay. 

52.  Bridge  or  Girder  Staying. — [Unwin,  §  763,  pages  138- 
141]. 

53.  Direct  Stays. — In  marine  boilers,  or  others  having  large 
diameter  and  relatively,  small   length,   the  portions  of   the   flat 


*  These  values  are  taken  from  Low's  Pocket  Book  for  Mechanical  Engi- 
neers. 


—  92  - 

heads  which  are  above  the  tubes  are  commonly  stayed  by  long 
bolts  passing  through  the  steam  space  from  one  head  to  the  other. 
These  bolts  have  nuts  with  washers  on  the  outside,  and  they 
should  be  spaced  so  that  each  of  the  different  bolts  support 
approximately  the  same  area  of  plate.  The  smallest  diameter  of 
the  stay  is  given  by  the  equation 


in  which  p  is  the  steam  pressure,  A  the  area  of  plate  supported 
by  the  stay,  and/  the  allowable  intensity  of  stress  in  the  stay. 
The  Board  of  Trade  allows  the  following  values  of  stress, 
/=5,ooo  pounds  per  sq.  inch  for  welded  wrought  iron  stays; 
f^  7,000  for  stays  of  wrought  iron  from  solid  bars  ;/=  9,000  for 
steel  stays  which  have  not  been  welded  or  worked  after  heating. 
The  greater  portion  of  the  heads  below  the  water  line  is  stayed 
by  the  tubes,  in  tubular  forms  of  boilers.  These  tubes  are  usually 
expanded  to  closely  fill  the  holes  bored  in  the  tube  sheets  or  heads  ; 
then  the  projecting  ends  of  the  tubes  are  "beaded",  or  riveted 
over.  The  tensile  stress  in  such  tubes  acting  as  stays  may  be 
taken  at  about  6,000  pounds  per  square  inch  of  cross-section  of 
metal  for  wrought  iron,  and  about  7,000  to  7,500  for  steel. 


VI. 
SCREWS,   BOLTS,  AND   KEYS. 


54.  Ordinary  uses  and  forms  of  Screws. — [Unwin,  §§  77, 
78,  pages  142  to  146.] 

The  triangular  section  threads  are  best  for  fastenings,  as  in  or- 
dinary screws,  studs,  and  bolts.  This  form  of  thread  has  more 
friction  between  the  threads  of  the  screw  and  the  nut  than  cor- 
responding square  threads,  thus  reducing  the  liability  to  unscrew. 
The  resistance  to  stripping  of  the  threads  is  also  greater  than  in 
"square  threads,"  for  similar  thickness  of  nuts.  On  the  other 
hand,  the  lower  frictional  resistance  of  the  square  thread  screw 
makes  this  form  suitable  for  transmission  of  energy.  The  angu- 
lar or  "  V  "  thread  has  one  advantage  over  the  square  thread  for 
such  cases  as  the  lead  screws  of  lathes  in  which  lost  motion  due 
to  wear  is  seriously  objectionable  ;  because  considerable  wear  of 
the  threads  can  be  taken  up  by  closing  the  split  clamp  nut ;  while 
lost  motion  in  the  true  square  thread  cannot  be  taken  up  in  this 
way.  To  obtain  this  great  practical  advantage  without  the  ex- 
cessive friction  of  "  V  "  threads,  an  intermediate  form  of  thread 
is  frequently  used  in  which  the  angle  between  the  sides  of  the 
threads  is  29°,  instead  of  60°  as  in  the  common  angular  thread. 
The  recognized  standard  screw  thread  in  the  United  States  is  the 
Sellers,  U.  S.,  or  Franklin  Institute  thread.  Consult  Fig.  86  (Un- 
win), and  the  table  on  page  94  of  these  Notes.  This  standard  is  not 
used  exclusively,  however,  but  a  full  "V"  thread  without  the 
flattened  tops  and  bottoms)  is  in  common  use.  The  angle  of  such 
"V"  threads  is  almost  always  60°  in  machine  bolts;  and  the 
number  of  threads  per  inch  usually  corresponds  to  those  of  the 
Seller's  system,  but  there  are  many  variations  in  this  particular. 

55.  Sellers,  or  United  States  Screw  Threads. — [Unwin,  § 
80,  pages  146-147.] 

The  pitch  of  screw,  (the  reciprocal  of  the  number  of  threads 


—  94  — 


per  inch)  is  the  same  in  both  the  Whitworth  and  the  Seller's  sys- 
tem for  all  sizes  below  iffe  inches  diameter,  except  for  ^2  inch  di- 
ameter. For  this  size  the  Whitworth  system  gives  12  threads 
per  inch,  while  the  Sellers  system  gives  13  threads  per  inch. 

SELLERS,  U.  S.,  OR  FRANKLIN  INSTITUTE  STANDARD  FOR  BOLTS. 


SCREW   THREADS. 

NUTS. 

BOLT   HEADS. 

d=  Out- 
side Di- 
ameter 
of 

Screw. 

N=  Num- 
ber of 
Threads  to 
an  Inch. 

ameter 
at  Root 
of 
Thread 
=  Diam. 
of  Hole 

Area  at 
Bottom 
Thread. 

Width  of 
Nut  Be 
tween  Par- 
allel Sides. 

T=  Thick- 
ness of 
Nut. 

w=  Width 
of  Head 
Between 
Parallel 
Sides. 

^Thick- 
ness of 
Head. 

in  Nut. 

Inches. 

Number. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

I 

2O 

18 
16 

.185 
.240 
•  294 

.027 

•045 
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The  Sellers  screws  have  much  greater  tensile  strength  than  full 
V  threads  of  equal  angles  and  pitch  ;  because  the  thread  of  the 
former  is  only  three  fourths  as  deep,  owing  to  the  flattening  at 
the  tops  and  bottoms.  The  depth  (/*)  of  a  full  V  thread  with  60° 
between  the  sides  is  equal  to  the  pitch  (p)  multiplied  by  the  co- 
sine of  30°  ;  or  .866  p.  Hence  if  d  is  the  outside  diameter  of  the 
screw  and  */,  is  the  diameter  at  the  bottom  of  the  thread, 

di  =  d  —  2/1  =  d  —  i.732/> 


—  95- 

for  the  full  depth  60°  thread.     The  depth  of  the  Sellers  thread  is 
Y±  X  .866/>^.65A      Hence  dl=  d—  1.30 p. 

The  area  ;it  bottom  of  a  i"  full  60°  thread  is  .482  square  inches  ; 
while  the  area  at  the  bottom  of  a  i"  Sellers  thread  is  .55  square 
inches,  or  14  per  cent.  more. 

56.  Machine  Screws  —The  small  sizes  of  screws,  with  slotted 
heads,  used  in  metal  work  are  (as  in  wire  and  sheet  metal)  usu- 
ally designated  by  numbers  rather  than  by  the  actual  diameter. 
But  in  the  standard  wire  gauges,  large  numbers  designate  small 
diameters,  while  in  machine  screws  large  numbers  indicate  large 
diameters.     The  following  formula  gives  the  actual   outside  di- 
ameters of  machine  screws  corresponding  to  the  number  by  which 
such  screws  are  designated, 

^^-.0131  N+  .057". 

This  number,  N,  is  simply  a  designation  of  the  size  and  is  not 
to  be  confused  with  the  number  of  threads  per  inch  (n).  The 
same  size  of  machine  screw  is  often  made  with  several  different 
numbers  of  threads  per  inch.  These  screws  are  usually  specified 
by  naming  the  size  number  first,  followed  by  the  number  of 
threads  per  inch.  Thus  :  an  18-20  machine  screw  means  size  18, 
and  20  threads  per  inch. 

57.  Pipe  Threads.— The  Briggs  system  of  Pipe  Threads  is  the 
established   standard    in    the    United    States.      The   numbers   of 
threads  per  inch  for  the  various  sizes  of  pipe  are  given  in  article 
26,  page  62,  of  these  Notes.     For  fuller  detail  see  Trans.  A.  S. 
M.  E.,  Vol.  VIII,  page  29. 

The  threads  given  in  §  79  (Unwin)  are  not  followed  in  the 
United  States. 

58.  Straining  Action  due  to  Load  Applied  to  Bolts.— The 
load  applied  to  bolts  is  generally  one  which  tends  to  separate  the 
connected  members,  and  this  action  is  resisted  by  a  tensile  stress 
in  the  bolts  ;  but  bolts  are  sometimes  used  to  prevent  the  relative 
translation  of  two  or  more  pieces,  when  a  shearing  stress  is  pro- 
duced in  the  bolts.     When  the  load  force  is  oblique  to  the  axis, 
the  stress  in  the  bolt  may  be  combined  tension  and  shearing.     If 


-96- 

any  screw  is  screwed  up  under  load  there  is  an  initial  direct  stress 
(tension  or  compression)  and  usually  a  torsional  stress  due  to  fric- 
tion between  the  threads  of  the  screw  and  the  nut.  With  bolts 
or  studs  screwed  up  hard,  as  in  making  a  steam  tight  joint,  the 
initial  tension  due  to  screwing  up  may  be  much  in  excess  of  that 
due  to  the  working  load.  This  will  be  treated  more  fully  later. 

If  the  load  applied  to  the  bolt  produces  a  shearing  action,  the 
bolt  shank  should  accurately  fit  the  holes  in  the  connected  pieces, 
at  least  for  the  portions  near  the  joint  ;  and  if  6"  is  the  load  per 
bolt,  a?  the  diameter  of  the  bolt  (shank),  and  /the  shearing  stress, 


In  a  bolt  subjected  to  a  lo  id  which  produces  tension,  the  mini- 
mum cross  section  sustains  the  greatest  stress.  This  smallest 
cross  section,  in  common  bolts,  is  through  the  bottoms  of  the 
threads 

If  /Ms  the  axial  load  carried  on  one  bolt,  /the  tensile  strength 
due  to  such  load,  and  d^  the  diameter  of  bolt  at  bottom  of  threads, 

/>=l4«/.-./=4_£ 

4  IT  a, 

See  Unwin,  §  82,  page  148. 

The  value  of  ^Trflf,  is  given  in  the  Table  on  page  94,  for  various 
sizes  of  Sellers  screws.  For  a  given  diameter  and  pitch  of  screw, 
the  area  at  the  bottom  of  threads  would  be  considerably  less  with 
full  "V"  threads. 

59.   Resilience  of  Bolts  with  Impulsive  Load. 

In  bridge  work  and  other  cases  requiring  long  bolts,  it  is  very 
common  to  make  the  cross  section  through  the  body  of  the  bolt 
about  equal  to  the  section  at  the  bottom  of  the  threads.  This 
may  be  done  by  upsetting  the  ends  where  the  thread  is  to  be  cut, 
or  by  welding  on  ends  made  from  stock  somewhat  larger  than 
that  used  for  the  main  length  of  the  bolt. 

The  most  apparent  result  of  this  practice  is  to  economize 
material  without  sacrifice  of  strength  (as  the  shank  still  has  an 
area  of  cross  section  equal  to  the  threaded  portion),  and  if  the 
weld  (when  the  ends  are  welded)  is  perfect,  the  strength  of  the 


—  97- 

bolt  is  not  reduced.  It  seems  probable  that  this  reason  is  re- 
sponsible for  the  original  adoption  of  this  practice,  since  it  has 
been  most  generally  used  in  long  tie  rods.  However,  in  case  of 
bolts  liable  to  shock,  there  is  an  even  more  important  reason  for 
such  construction  ;  since  it  can  be  shown  that  the  reduced  section 
not  only  maintains  the  full  strength  under  static  load,  but  it  very 
greatly  increases  the  capacity  of  the  bolt  to  resist  shock.  This 
last  fact  has  not  been  very  generally  recognized,  as  appears  from 
the  common  application  of  such  reduced  shank  bolts  only  to 
structures,  rather  than  to  machines. 

It  has  been  seen  that  the  resistance  of  a  tension  member  under 
a  static  load  is  determined  solely  by  its  weakest  section  ;  while, 
in  a  member  subjected  to  shock,  impact,  or  impulsive  load,  the 
resistance  depends  upon  the  total  extent  of  distortion  of  the  mem- 
ber due  to  a  given  intensity  of  stress. 

As  shown  in  art.  8,  the  maximum  stress  with  impulsive  load  is 
^  _  W(h  +  /) 
KIA 

For  a  stress  within  the  E.L, 


This  shows  clearly  that  for  a  given  load,  IV,  applied  suddenly 
or  with  impact,  the  stress  produced  in  a  member  of  sectional  area, 
A,  is  greater  as  /  becomes  less  relative  to  h.  Hence,  if/  is  in- 
creased, the  stress  produced  becomes  less  for  a  given  impulsive 
action  ;  or  the  resistance  to  such  action  is  greater  for  a  given 
value  of  the  stress. 

If  an  ordinary  bolt  is  subjected  to  shock  in  a  direction  to  pro- 
duce tension,  the  stress  will  be  a  maximum  at  the  sections 
through  the  bottom  of  the  threads  ;  the  bolt  will  elongate  but 
the  elongation  will  be  confined  largely  to  the  very  short  reduced 
(threaded)  sections,  hence  the  stress  will  be  much  less  in  the 
larger  portion  of  the  bolt.  In  a  Sellers  bolt  of  one  inch  diame- 
ter the  area,  A,  of  the  shank  is  .ySsq.  inches,  while  the  area,  A', 
at  the  bottom  of  threads  is  only  .55  sq.  inches.  Therefore  a 
7 


-98- 


stress  on  A'  of  30000  Ibs.  per  sq.  in.  =  _i      —  2IOOO    on 

.78 

the  full  sections.  Suppose  the  elongation  per  inch  of  length  at  a 
stress  of  30000  (taken  as  the  E.  L,.)  is  TsW-  Each  inch  of  section 
A'  will  elongate  roW'-  while  each  inch  of  full  sectional  A(=.jS  sq. 
in.)  will  have  a  stress  of  only  21000  Ibs.,  with  a  corresponding  elon- 
gation of  |-J  X  TinsV  ~  -0007".  Assume  the  thread  to  be  i"  long, 
and  the  remainder  of  the  bolt  to  be  5"  long.  It  will  appear  that 
the  mean  stress  on  the  threaded  portion  (i")  is  about  the  mean 
of  30000  and  21000,  or  say  25500  Ibs.  per  square  inch  ;  as  the 
mean  section  is  an  average  of  .55  and  .78  square  inches.  Hence 
the  elongation  for  this  threaded  r  inch,  when  the  stress  on 
A'  =  30000,  is  .00085",  while  the  other  5"  (of  area  A}  will  elon- 
gate under  this  load  5  X  .0007  =  .0035".  The  total  elongation 
will  then  be  /=  .00085  +  -0035  =  .00  135  inches. 

**=  -^  ^r 

•55X30000  x  .00435  =  8250  x          6  =         lbs< 

2  -10435 

Now  suppose  the  5"  shank  of  this  bolt  were  reduced  in  section 
to  an  area  A'  =  .55''.     Then  the  elongation  of  this  portion  under 
the  above  load  would  be,  5  X  .001  =  .005'',  instead  of  .0035"  and 
the  total  elongation  would  be  /=  .00085  +  -°°5  —  -00585. 
4<   w=  .55  X  30000  x  .00585  = 

2  .  10585 

This  latter  load  is  33  per  cent,  greater  than  the  preceding. 
With  a  "V"  thread,  not  reduced  in  the  shank  the  case  would  be 
much  worse,  as  the  reduced  section  is  very  small  in  length,  theo- 
retically it  is  zero. 

The  preceding  example  shows  that  the  elastic  resilience  of  the 
bolt  was  increased  33  per  cent,  by  reducing  the  body  of  the  bolt 
to  A'  '.  Of  course  the  gain  would  be  still  greater  with  a  longer 
bolt.  It  may  be  well  to  remember  that  the  "  long  specimen  "  is 
more  apt  to  contain  a  weak  section  than  is  a  short  specimen  ; 
but,  on  the  other  hand,  the  sharp  notching  of  the  threads  is  quite 
liable  to  start  a  fracture  at  their  roots. 


—  99  — 

If  the  bolt  is  strained  beyond  the  elastic  limit,  the  portion  thus 
strained  yields  at  a  much  greater  rate,  relative  to  the  stress,  than 
that  given  above.  With  a  load  which  would  produce  a  stress  of 
30000  Ibs.  per  sq.  in.  in  the  larger  portion  (area  A}^  the  stress  in 

the  reduced  portion  (area  A')  will  be^     ><^185  =  43200   Ibs. 

per  sq.  inch.  Hence,  the  effect  of  a  long  section  in  resisting 
shock  ivithout  rupture  is  much  greater  even  than  that  shown  for 
elastic  deformation  only. 

The  section  of  the  shank  of  the  bolt  may  be  reduced  as  in  Fig. 
35,  by  turning  down  the  body  of  the  bolt  to  about  the  diameter 
at  the  bottoms  of  the  threads.  The  collars  a  and  a  may  be  left 
to  form  a,  fit  in  the  hole.  This  form  is  easy  to  make,  but  does  not 
fit  the  hole  throughout  its  length,  and  it  is  weak  in  torsion. 

Fig.  36  is  somewhat  more  expensive,  but  fits  the  hole  better 
and  is  somewhat  stronger  in  torsion.  Fig.  37  is  the  form  which 
gives  the  best  fit,  and  is  also  the  strongest  in  torsion.  If  very 
long  it  is  difficult  to  make  ;  otherwise  it  is  perhaps  the  best. 

These  high  resilience  bolts  only  increase  the  resistance  to  im- 
pulsive load,  not  to  dead  load.  They  are  good  forms  to  use  in 
such  cases  as  the  so-called  "marine  type"  of  connecting  rod, 
where  the  bolts  are  subjected  to  considerable  shock. 

For  cylinder  head  bolts,  and  other  cases  where  a  tight  joint  is 
the  main  consideration,  this  form  of  bolt  may  be  entirely  unsuited. 

Professor  Sweet  prepared,  for  tests,  some  bolts  such  as  are 
used  in  the  connecting  rod  of  the  Straight  Line  Engine  ;  of  these, 
half  were  solid  (ordinary  form)  bolts,  and  the  other  half  were  of 
the  form  shown  in  Fig.  37. 

Tests  of  a  pair  of  these  bolts,  one  of  each  kind,  showed  an  elon- 
gation at  rupture  of  .25"  for  the  solid  bolt,  which  broke  in  the 
thread;  while  the  drilled  bolt  elongated  2.25",  or  9  times  as 
much,  and  it  broke  through  the  shank,  the  net  section  of  which 
was  a  trifle  less  than  that  at  the  bottom  of  the  threads.  Drop 
tests  showed  similar  results.  These  tests  indicate  the  superior 
ultimate  resilience  of  the  reduced  shank  bolts. 

60    Friction  and   Efficiency  of  Screws  and  Nuts.— When 


a  bolt  is  screwed  up  under  load  a  torsional  stress  is  produced  in 
it,  due  to  the  factional  resistance  overcome  at  the  threads.  If.  in 
screwing  up  the  bolt,  pressure  is  produced  between  the  members 
connected,  their  reaction  may  cause  a  considerable  initial  tension 
in  the  bolt ;  in  fact,  this  initial  tension  due  to  screwing  up  is  fre- 
quently much  greater  than  that  due  to  the  external  ("useful  ") 
load.  The  above  mentioned  stresses  are  much  affected  by  the 
friction  of  the  threads  and  of  the  nut  on  its  seat ;  for  this  reason 
the  friction  of  screws  is  considered  at  this  point.  Read  Umvin, 
§  83,  as  far  as  the  bottom  of  page  149. 

Referring  to  Fig.  87  (Unwin),  it  is  to  be  noted  that  the  force  Q 
is  due  to  the  pull  on  the  wrench.  Of  this  pull,  one  portion  is  ex- 
pended in  overcoming  the  friction  of  the  nut  on  its  seat,  while  the 
remainder,  reduced  to  the  equivalent  force  acting  at  the  mean 
radius  of  the  thread,  is  this  force  Q.  The  actual  pull  on  the 
wrench  (neglecting  friction  of  the  nut  on  the  seat)  is  Q  multiplied 
by  the  mean  radius  of  the  b,olt  and  divided  by  the  effective  radius 
of  the  wrench. 

From  the  consideration  that  the  "  input  "  of  energy  must  equal 
the  "  output  "  (including  frictional  losses),  the  following  relations 
are  deduced  from  examination  of  Fig.  87  (Unwin)  : 

Q  -  bc-^P  •  ac+  F  •  ab,  (i) 

or  Q  cos  a  =  P  sin  a  +  F,  ( i') 

because  b  c  :  a  c  :  a  b  : :  cos  a  :  sin  a  :  i , 

in  which  a  =  the  inclination  of  the  screw  threads  —  angle  a  be. 
Also,  the  friction  equals  the  normal  pressure  between  the  sliding 
surfaces  multiplied  by  the  coefficient  of  friction,  or  F=  R p.. 
From  the  relation  that  the  sum  of  the  vertical  forces  equals  zeio, 
P=  R  cos  a  —  .Fsin  a  =  R  (cos  a  —  ft.  sin  a)  (2) 

--.    R  = —  (3) 

COS  a  —  (J,  Sill  a 

Equation  (i')  may  now  be  reduced  to  the  following  : 
Q  cos  o  —  /'sin  a  +  R  (JL  =  Psin  a+ *_* — 

COS  a  —  /A  Sill  a 

=  p  pin  a  COS  a  +  /*(l  —  SJn2a)  "i  —  pCQ5  a  (  sin  a  +  /*  COS  a  "I 
L  COS  a  —  /iSina  J  '      LcOSa  —  /*  sill  a  J 

(4) 


( 
a  —  *ju,-sin  aj 

Since  the  circumference  of  the  screw  (TflO  is  to  the  pitch  (^) 
as  be  is  to  a  c<  or  as  cos  a  :  sin  a,  equation  (5)  may  be  written  thus, 


TT  d  —  p.p  J 

This  last  expression  is  eq.  (5)  of  Unwin,  §  83. 

The  coefficient  of  friction,  yu.,  is  equal  to  the  tangent  of  the  angle 
of  repose  (<£)  of  the  two  surfaces  in  contact,  or  ^  =  tan  </>.  Divid- 
ing numerator  and  denominator  in  eq.  (5)  by  cos  a,  and  putting 
in  tan  <f>  for  /x, 


<M       p  ^  ( 
<>) 


vi  —tan  a  tan  <j> 
Inspection  of  eq.  (Y)  shows  that  for  a  frictionless  screw  (F=  o), 

£>cosa=  /'sin  a  or  Q  =  P  S1"  a=  P  tan  a  ; 

COS  a 

while  eq    (6)  shows  that  with  friction  Q  =  Plan  (a  +  <£). 

Hence  the  effect  of  friction  is  to  require  an  expenditure  of  ener- 
gy equivalent  to  screwing  up  a  frictionless  bolt  with  threads  in- 
clined at  an  angle  (a  +  <£)  ;  while  the  useful  work  actually  per- 
formed with  this  expenditure  of  energy  is  only  that  due  to  a  screw 
of  inclination  a. 

With  triangular  thread  screws,  the  normal  pressure  at  the 
threads  is  greater  than  with  square  threads  ;  hence  the  friction  at 
the  threads  is  greater,  other  things  being  equal.  In  Fig.  38-  the 
normal  pressure  for  a  square  thread  is  indicated  by  R,  while  the 
resultant  normal  pressure  for  triangular  thread  is  ^"=./?sec0, 
in  which  0  =  half  the  angle  between  the  adjacent  faces  of  a 
thread.  R"  represents  the  radial  crushing  action  on  the  thread 
of  the  screw,  and  its  equal  and  opposite  reaction  tends  to  burst 
the  nut.  With  60°  angular  thread,  as  in  the  Sellers'  system,  or 
the  common  "V"  thread,  R'  =  R  sec  30°  =  1.15^?.  The  fric- 
tion is  increased  directly  as  the  normal  pressure  ;  or  it  is  about  15 
per  cent,  greater  in  the  60°  angular  thread  than  in  the  square 
thread. 


As  the  friction,  /%  is  R  p.  sec  6  for  triangular  threads,  equation 
(2)  may  be  written  thus  • 

P=RcQsa  —  ^? /A sec ^ sin  a  ..-.  R=—         —£—  (7) 

cos  a  —  p.  sec  6  sin  a 

Kquation  (i')  then  gives, 

Ocosa  =  Ps'm  a  +  R  u.  sec  6  =  Ps'm  a  -f £L- 

cos  a  —  ju.sec0sina 

oj  sin  a  cos  a  -f  /xsecfl  (i  —  sin2  a)  "I 

cos  a  —  fj.  sec  0  sin  a  J 

=  />cos a 


_cos  a  —  p.  sec  0  sin  ay 

L    -f 

t    - 

Or,  when  0  =  30°, 


~       D  i  run  a -f  jLisec0cosa^_   D/'/H-  p-Trdsec  B~\      ,„•. 
'•Q=^\~  ._/»...     \—p\^ —          ^        ^8^ 


This  last  is  the  same  as  eq.  (6)  of  Unwin,  page  150.  The  por- 
tion of  §  83  (Unwin)  below  eq.  (6)  may  be  omitted. 

When  the  friction  of  the  nut  on  its  seat  (or,  of  the  thrust  collar 
against  its  bearing)  is  considered,  the  force  Q  is  the  entire  turn- 
ing force  reduced  to  its  equivalent  acting  at  the  mean  radius  of 
the  threads.  Hence  for  square  threads, 


VTT  d—  p.pj 

in  which  dl  is  the  mean  friction  diameter  of  the  nut  or  collar)  and 
^  is  the  coefficient  of  friction  of  this  nut  on  its  seat.  If  the  ratio 
of  d^  to  d  be  called  a, 

Q=p(t+^+a^\=p(*»«  +  *co*a+a^     (10) 
\.ird  —  p.p  l-COSa  —  /A  sill  a 

For  triangular  threads,  terms  containing  /A  in  eq.  (9)  should  be 
multiplied  by  the  secant  of  half  the  angle  between  adjacent  faces 
of  the  thread,  or  by  sec  0,  as  in  eqs.  (S)  and  (9)  above  ;  but  the 
term  containing  tij  is  not  to  be  so  multiplied  because  the  friction 
of  the  nut  on  the  seat  is  not  affected  by  the  form  of  the  thread. 

For  the  ordinary  standard  forms  of  nuts  the  mean  friction  diam- 


-  103- 

eter  of  the  nut  may  be  assumed  at  about  i^  times  the  diameter  of 
the  bolt;  or  d^  =  ^d,  hence,  in  eq.  (9),  the  coefficient  a  =  -f. 
With  the  Sellers'  system  of  threads,  or  the  most  usual  form  of  full 
"V"  threads,  the  normal  pressure  at  the  threads  is  1.15  times 
the  pressure  for  square  threads,  as  already  noted. 

Dividing  both  numerator  and  denominator  of  the  fractional  part 
of  eq.  (10)  by  cos  a,  and  inserting  the  values  of  a  and  of  sec  0  as 
just  assigned,  for  standard  fccrew  threads, 


Vi  —  1.  15  /A  tan  a 
In  the  Sellers'  system,  a  varies  from  about  2°  45'  in   a  ^  inch 
screw  to    i°   45'  in  a  3  inch  screw  ;  or  tan   a  varies  from  .049  to 
.0303  in  this  same  range.     If  /x  be  taken  at  .  15  and  /*,  at  .  10  it  ap- 
pears that  Q  varies  from  .356  P  with  a  •£"  screw  to  .337  P  with  a 
3"  screw.     The  coefficients  of  friction  will  vary  much  more  than 
this,  so  it  may  be  assumed  that  for  the  ordinary  range  of  bolts, 
Q=  .345  P  (approximately.)  (12) 

If  friction  could  be  entirely  eliminated  eq.  (i)  would  become 
Q-ab  =  P-ac+o\  or  Q  IT  d  =  P  p 


IT  d 

Then  for  a  standard  i  inch  bolt  with  no  friction,  Q  =  .04  P  ;  while 
with  the  above  assumptions  as  to  friction,  Q=  .345  P,  approxi- 
mately. The  ratio  of  these  two  values  of  Q  gives  an  expression  for 
the  efficiency  of  the  screw  and  nut,  and  for  the  above  conditions, 
this  is  about  11.5  per  cent.  It  will  be  seen  in  the  next  article 
that  this  result  agrees  quite  closely  with  certain  direct  experi- 
ments. 

The  relation  between  the  load  on  a  screw,  />,  and  the  tangential 
turning  force,  Q,  is  given  by  eq.  (5)  of  this  article,  when  the  screw 
is  being  turned  so  as  to  produce  a  motion  of  the  loaded  member 
opposite  to  the  direction  of  the  force  P.  This  direction  of  motion 
will  be  designated  as  "  hoisting,"  though  the  load  may  not  be  actu- 
ally moved  upward  by  this  action.  With  the  usual  proportions, 
the  load  will  not  drive  the  screw  backward  when  the  turning 


-  104  — 

force  Q  ceases  to  act,  i.  e.,  the  screw  mechanism  will  not  "over- 
haul ;  "  but  some  force  opposite  in  direction  to  Q  must  be  applied 
to  "  lower."  Screws  may  be  so  proportioned  that  they  will  over- 
haul, but  this  condition  is  not  usual. 

Referring  again  to  Fig.  87,  Unwin,  it  is  apparent  that  if  the 
load  is  being  lowered  the  friction,  /•*,  would  be  opposite  in  direc- 
tion to  that  indicated,  as  the  friction  always  opposes  the  relative 
motion  of  the  members  between  which  it  acts.  It  will  be  as- 
sumed that  the  direction  of  Q  is  as  yet  unknown  but  its  direction 
in  hoisting  (that  indicated  in  Fig.  87)  will  be  called  positive. 
As  stated  above,  with  the  usual  proportions  of  sciews  Q  is  nega- 
tive. The  equation  for  lowering,  which  corresponds  to  eq.  (i) 
of  this  article,  becomes 

Q-bc=P-ac—  F-ab  (13) 

or  Q  cos  a  =  Psin  a  —  F  (I3/) 

in  which  it  remains  to  be  seen  whether  Q  is  positive  or  negative. 
It  is  evident,  however,  that  the  work  P  at  which  measures  the 
tendency  of  the  load  to  run  down  is  opposite  in  sign  to  F •  ab 
which  resists  overhauling.  As  before,  F=Rp.^  and  from  the 
equality  of  the  vertical  components  of  the  concurrent  forces, 

P '=--  R  cos  a  -\-  F 'sin  a  =  R  (cos  a  -j-  /A sin  a)  (14) 

From  eqs.  (13')  and  (14) 

Qcosa=Ps\na  —  Rf*.=  P\  sin  a— £ — : —  J 

\  COS  a  -(-  /A  Sill  a/ 


MX>Sa  -f  /A  Sill  a  7 

To  investigate  the  sign  of  Q,  assume  that  it  equals  zero,  as  it 
will  for  a  certain  relation  of  a  and  /A.     Then 

a  —  a  COS  a  \ 

1  —  ~    . '.  Sill  a  =  fj.  COS  a 


COS  a.  -f-  ft.  sin  a/ 

-a  =  tan  a  =  a  =  tan  <£  (16) 

cos  a 

since  the  coefficient  of  friction   (/A)    equals   the   tangent   of  the 
angle  of  repose  (</>).      Hence,  when  £?  =  o,  «  =  <i>,  or  the  inclina- 


tion  of  the  thread  helix  equals  the  angle  of  repose.  The  angle 
of  repose  depends  upon  the  nature  of  the  materials  used,  their 
condition  as  to  finish,  and  the  lubrication. 

If  a  ^>  <f>,  tan  a  ^>  p.  .'.  sin  a  >  /icosa  (multiplying  both  sides 
of  the  inequality  by  cos  a);  hence  the  numerator  of  eq.  (15) 
(sin  a  —  /A  cos  a)  is  positive,  and  Q  is  positive.  It  is  evident  that 
this  must  be  so,  for  if  Q  =  o  when  a  =  <f>,  Q  must  be  positive  for 
an  inclination  of  the  screw  greater  than  the  angle  of  repose  to 
prevent  overhauling. 

If  a  <^  <£,  tan  a  <^  /x  .  '.  sin  a  <^  /*  cos  a;  hence  sin  a  —  p.  cos  a  is 
negative  and  Q  must  be  negative  ;  or  the  force  P  will  not  over 
haul  the  mechanism  when  the  inclination  of  the  helix  is  less 
than  the  angle  of  repose,  and  some  force  (—  Q)  must  assist  the 
tendency  of  P  to  overhaul  before  lowering  can  actually  occur. 

Since  Q  is  usually  negative,  it  is  convenient  to  write  Q=  —  Q, 
when  eq.  (15)  reduces  to 


fji  sin 
Or,  substituting  p  for  sin  a,  and  ir  d  for  cos  a, 

(  0 


The  friction  of  the  nut  or  thrust  collar  on  its  seat  increases 
work  to  be  overcome  in  lowering  as  well  as  in  hoisting,  and  it 
is  to  be  added  to  eq.  (17')  above.  Including  this  resistance 


in  which  a  is  the  ratio  of  mean  friction  diameter  of  the  nut  (or 
collar)  to  the  mean  diameter  of  the  screw  threads,  and  /u.,  is  the 
coefficient  of  friction  of  the  nut  on  its  seat,  as  before. 

For  a  "  V  "  thread  /*  should  be  multiplied  by  the  secant  of  half 
the  angle  between  the  adjacent  surfaces  of  the  threads  ;  but  as  the 
thread  most  commonly  used  for  transmission  of  energy  is  either 
a  square  thread,  or  one  approximating  it,  equation  (18)  of  this 
article  will  ordinarily  apply  in  computations  relating  to  lowering. 

61.   Initial  Tension  in  Bolts  due  to  Screwing  Up.    [Unwin, 


—  io6  — 

§85,  page  151,]  It  is  assumed  by  Unwin  that  the  radius  of 
wrench  will  usually  be  about  15  times  the  diameter  of  the  bolt, 
and  that  the  heaviest  ordinary  pull  of  the  workman  will  be 
about  30  Ibs.  On  this  basis,  he  estimates  that  the  initial  tension 
on  a  bolt  due  to  screwing  up  will  be  about  2500  Ibs.,  regardless 
of  the  size  of  the  bolt ;  although  it  is  stated  in  this  connection 
that  experience  teaches  the  mechanic  in  what  case  a  heavy  pres- 
sure may  be  applied  with  .safety.  While  this  view  seems  plausi- 
ble, it  is  probable  that  the  initial  tension  due  to  screwing  a  nut 
up  tight  is  usually  very  much  greater  than  2500  Ibs. 

A  series  of  experiments  was  made  in  the  Sibley  College  Labo- 
ratory, a  few  years  ago,  to  directly  determine  the  probable  load 
produced  in  standard  bolts  when  making  a  tight  joint.  The 
sizes  of  bolts  used  were  £",  f",  i"  and  i\" '.  One  set  of  experi- 
ments was  made  with  rough  nuts  and  washers,  and  another  set 
with  the  nuts  and  their  seats  on  the  washers  faced  off.  A  bolt 
was  placed  in  a  testing  machine,  so  that  the  axial  force  upon  it 
could  be  weighed  after  it  was  screwed  up.  Each  of  twelve  expe- 
rienced mechanics  was  asked  to  select  his  own  wrench  and  then 
to  screw  up  the  nut  as  if  making  a  steam-tight  joint,  and  the 
resulting  load  on  the  bolt  was  weighed.  Each  man  repeated  the 
test  three  times  for  every  size  of  bolt,  and  each  had  a  helper  on 
the  i"  and  i£"  sizes.  The  sizes  of  wrenches  used  were  10"  or  12" 
on  the  %"  bolts  up  to  18"  and  22"  on  the  i^"  bolts.  The  results 
were  rather  discordant,  as  should  be  expected  ;  the  loads  in  the 
different  tests  were  rather  more  uniform,  as  well  as  higher,  with 
the  faced  nuts  and  washers.  The  general  result  indicates  :  (a) 
that  the  initial  load  due  to  screwing  up  for  a  tight  joint  varies 
about  as  the  diameter  of  the  bolt  ;  that  is,  a  mechanic  will  grad- 
uate the  pull  on  the  wrench  in  about  that  ratio,  (b)  That  the 
load  produced  may  be  estimated  at  16,000  Ibs.  per  inch  of  diame- 
ter of  bolt,  or 

/>  =  16,000  </  (i) 

in  which  Pl  is  the  initial  load  in  pounds  due  to  screwing  up,  and 
d  is  the  nominal  (outside)  diameter  of  the  screw  thread.  This 
value  of  Pl  is  rather  above  the  average  for  the  tests  ;  but  it  is  con- 


-  107  - 

siderably  below  the  maximum,  and  it  is  probably  not  in  excess  of 
the  load  which  may  reasonably  be  expected  in  making  a  tight 
joint. 

If  the  initial  load  due  to  screwing  up  be  divided  by  the  cross- 
sectional  area  of  the  bolt  at  the  bottom  of  the  threads,  the  initial 
intensity  of  the  tensile  stress  is  obtained.  The  above  experiments 
indicate  that  this  intensity  of  stress  varies,  approximately,  inverse- 
ly as  the  nominal  diameter  (d~)  of  the  bolt ;  and  that  it  may  fre- 
quently equal  or  exceed 

/=  3Q.OOQ  lbs   persq.  in.  (2) 

a 

In  addition  to  this  tensile  stress  there  is  a  considerable  twisting 
action  on  the  bolt.  Equation  (2)  would  give  a  stress  of  60,000 
Ibs.  per  square  inch  on  a  ^  inch  bolt  ;  and  this  result  is  sub- 
stantiated by  the  fact  that  steel  bolts  of  this  size  were  broken  in 
the  course  of  the  experiments.  It  also  agrees  with  common  ex- 
perience which  forbids  the  use  of  screws  as  small  as  ^2  inch  for 
cases  requiring  the  nuts  to  be  screwed  up  hard. 

In  these  experiments,  the  average  effective  lever  arm  of  the 
wrench  was  not  far  from  15  times  the  diameter,  or  30  times  the 
radius,  of  the  screw  ;  hence,  if  it  be  assumed  that  the  turning 
force  acting  at  the  radius  of  the  screw  is  Q  =  .34.5  P,  as  in  eq. 
{12)  of  art.  60,  the  force  applied  at  the  wrench  is,  in  pounds, 
about 

Q^  &-  =  '345/?  =  -345X  16.000  d_  igo  d 

30  30  30 

instead  of  30  Ibs  for  all  sizes  of  bolts,  as  assumed  by  Unwin. 

Unwin's  assumption  would  indicate  that  the  intensity  of  stress 
varies  inversely  as  the  square  of  the  diameter,  instead  of  inversely 
as  the  first  power  of  the  diameter.  On  the  other  hand,  a  common 
practice  in  designing  is  to  take  a  low  working  stress  (large  factor 
of  safety)  "to  allow  for  the  stress  due  to  screwing  up.''  This 
procedure  implies  that  the  intensity  of  initial  stress  is  the  same 
for  all  sizes  of  screws,  which  does  not  seem  to  be  justified  either 
by  reason  or  experience. 

The  above  discussion  indicates  that  the  factor  of  safety  should 


—  io8  -— 

be  increased  as  the  size  of  screw  decreases,  and  of  course  this 
factor  should  be  varied  with  the  conditions  of  the  case,  as  in 
some  applications  the  nuts  are  much  more  apt  to  be  screwed  up 
hard  than  in  others. 

A  set  of  experiments  were  made  by  Mr.  James  McBride 
(Trans.  A.  S.  M.  E.,  Vol.  XII,  page  781)  which  show  that  the 
factor  of  safety,  as  bolts  are  frequently  used,  is  very  low,  even 
with  a  very  moderate  external  load.  One  case  cited  by  Mr. 
McBride  indicates  that  the  stress  due  to  screwing  up  a  3^6  inch 
bolt  was  nearly  one-half  the  ultimate  strength,  or  probably  very 
near  the  elastic  limit,  His  direct  determinations  of  the  efficiency 
of  a  standard  2  inch  screw  bolt  shows  an  average  of  only  10.19 
per  cent.  It  is  probably  this  low  efficiency  which  saves  many 
screws  from  being  broken,  as  the  frictional  loss  reduces  the 
tension  produced  in  the  bolt  by  screwing  up.  The  excessive 
friction  makes  the  screw  bolt  a  useful  fastening,  as  it  reduces  the 
tendency  to  "overhaul  "  or  unscrew. 

62  Resultant  Stress  on  Bolts  due  to  Combined  Initial 
Tension  and  External  Load. — It  was  shown,  in  article  61,  that 
bolts  may  be  subjected  to  a  high  tensile  stress  by  screwing  up  the 
nuts.  The  question  often  arises  as  to  the  effect  of  the  combined 
action  of  this  initial  tension  and  the  external,  or  useful,  load.  It 
is  stated  by  some  that  the  resultant  load  on  the  bolt  is  simply  the 
sum  of  the  initial  and  the  external  loads  Others  contend  that 
the  application  of  the  external  load  does  not  change  the  stress  in 
the  bolt,  unless  this  external  load  exceeds  the  initial  load  due  to 
screwing  up  ;  that  is,  that  the  resultant  load  is  equal  to  either  the 
initial  load  alone,  or  to  the  external  load  alone,  whichever  is  the 
greater. 

Neither  of  these  views  is  entirely  correct  for  conditions  at- 
tained in  practice.  They  represent  the  extreme  limiting  cases 
and  every  actual  case,  lies  between  them. 

If  the  bolt  itself  could  be  absolutely  rigid  while  the  members 
forced  together  in  screwing  it  up  yielded  under  pressure,  the  total 
load  on  the  bolt  would  be  equal  to  the  sum  of  the  initial  load  and 
the  external  load.  If,  however,  the  members  pressed  together 


were  absolutely  rigid,  only  the  bolt  yielding,  the  total  (resultant) 
load  on  the  bolt  would  be  the  initial  load  alone,  or  the  external 
load  alone,  whichever  is  the  greater. 

The  first  of  the  above  conditions  is  approached  by  the  arrange- 
ment shown  in  Fig.  39.  Screwing  up  the  nut  compresses  the 
spring  interposed  between  A  and  B.  Assume  that  an  axial  force 
of  2000  pounds  will  compress  this  spring  i  inch  ;  then  if  the  nut 
is  screwed  up  till  the  spring  is  2  inches  shorter  than  its  free  length, 
the  load  on  the  bolt,  due  to  screwing  up,  must  equal  the  reaction 
of  the  spring,  or  4000  Ib-;.  Assume,  also,  that  the  extension  of 
the  bolt  under  this  screwing  up  action,  or  under  the  initial  load 
of  4000  Ibs.,  is  .02  inch.  Now,  if  an  external  axial  load  of  say 
2000  Ibs.  be  applied  to  the  eye  at  the  bottom  of  B,  this  added  load 
would  tend  to  further  increase  the  length  of  the  bolt  by  about  .01 
inch  ;  but  this  further  extension  of  the  bolt  would  reduce  the 
compression  on  the  spring  by  a  corresponding  amount  and  thus 
slightly  diminish  the  spring  reaction.  With  such  great  differ- 
ence between  the  rigidity  of  the  bolt  and  of  the  connected  mem- 
bers, the  load  on  the  bolt  becomes  practically  the  sum  of  the  ini- 
tial and  the  external  loads,  but  the  resultant  load  is  necessarily 
somewhat  less  than  this  sum  in  any  possible  case. 

The  arrangement  shown  in  Fig.  40  is  one  which  approaches 
the  other  limiting  case  mentioned  above.  Suppose  the  bolt  to 
be  a  spring  which 'is  subjected  to  an  axial  load  of  4,000  Ibs.  in 
screwing  the  nut  up  2  inches,  and  that  the  corresponding  yield- 
ing of  the  member  B  is  .02  inch.  The  initial  load  on  the  bolt 
(which  is  the  spring  in  this  case)  is  4,000  Ibs.,  and  the  pressure 
between  the  contact  surfaces  of  A  and  B  is  equal  to  it.  If  an  ex- 
ternal axial  load  be  now  applied  to  the  eye  in  B,  the  pressure  be- 
tween the  contact  surfaces  is  reduced  by  an  amount  nearly  equal 
to  this  external  load.  But  unless  the  external  load  exceeds  the 
initial  load,  the  bolt  will  not  elongate  enough  to  separate  these 
contact  surfaces  and  entirely  remove  the  pressure  between  them, 
because  the  load  on  the  bolt  (spring)  cannot  change  without 
changing  the  length  of  the  bolt,  and  with  the  above  data  the 
bolt  would  have  to  stretch  an  additional  .02  inch  (equal  to  the 


initial  yielding  of  the  connected  members)  before  the  contact  sur- 
faces would  be  entirely  relieved  of  pressure.  It  therefore  appears 
that  the  addition  of  an  external  load  in  this  case  does  not  materi- 
ally affect  the  resultant  tension  on  the  bolt  as  long  as  this  external 
load  does  not  exceed  the  initial  load.  If  the  external  load  is 
greater  than  the  initial  load  (say  6,000  Ibs.)  the  elongation  of  the 
bolt  increases  (to  3  inches)  ;  the  resultant  load  on  the  bolt  will  be 
simply  the  external  load  alone,  because  the  latter  is  sufficient  to 
entirely  relieve  the  pressure  produced  between  the  contact  surfaces 
in  screwing  up. 

In  all  ordinary  practical  cases  the  difference  in  rigidity  between 
the  bolt  and  the  connected  members  is  much  less  than  in  the  ex- 
treme conditions  considered  above.  The  resultant  load  on  a  bolt 
may  be  anything  between  the  sum  of  the  initial  and  the  external 
loads  as  a  maximum,  and  the  greater  of  these  two  loads  alone  as  a 
minimum.  This  resultant  load  approaches  the  maximum  limit 
when  the  bolts  are  rigid  relative  to  the  connected  members  as  in 
Fig.  41  ;  and  this  resultant  approaches  the  minimum  limit  when 
the  bolts  are  relatively  yielding,  as  in  Fig.  42.  In  any  particular 
case  the  designer  can  tell  which  limit  is  the  more  nearly  ap- 
proached, and  he  should  be  governed  accordingly. 

The  "Locomotive"  (Nov.,  1897)  contains  an  excellent  article 
on  the  resultant  load  on  bolts,  and  a  relation  is  derived  from 
which  the  following  method  of  treatment  has  been  developed  : 
The  application  of  this  method  depends  simply  upon  the  ratio  of 
the  yield  of  the  connected  members  to  the  yield  of  the  bolts.  It 
will  usually  not  be  difficult  to  assign  a  sufficiently  close  value  to 
this  ratio,  even  when  the  actual  magnitudes  of  yielding  are  un- 
known ;  in  fact,  only  a  rough  approximation  to  the  value  of  this 

V 

ratio  is  necessary.     Let  this  ratio  be  called  y  and  let  =  x  ; 

call  the  initial  load  on  the  bolt  due  to  screwing  up  P^ ;  the  ex- 
ternal (useful)  load  Pt ;  and  the  total  (resultant)  load  P.  Then 
it  can  be  shown  that 

P=Pl  +  xP^.  (i) 

If  the  yield  ratio  (jv)  is  known,  the  value  of  x  is  at  once  found  by 


the  above  relation  of  x  audjy.  If  the  yield  of  the  connected  mem- 
bers is  between  i  and  5  times  that  of  the  bolt,  the  resultant  load 
is  equal  to  the  initial  load  added  to  from  0.5  to  0.8,  the  external 
load.  If  a  tight  joint  it  made  with  short  rigid  bolts  or  studs,  con- 
necting flanges  which  are  separated  by  an  elastic  packing,  or  with 
a  metal  contact  at  some  distance  from  the  centre  line  of  the  bolts, 
as  indicated  in  Fig.  41,  the  applied  load  is  an  important  consider- 
ation since  the  value  of  y  is  relatively  great..  In  some  other  cases 
the  external  load  may  be  a  minor  consideration  as  affecting  the 
strength  of  the  bolt. 

When  the  conditions  are  such  that  the  nut  is  not  apt  to  be 
screwed  up  hard,  that  is  when  the  initial  load  may  be  safely 
neglected,  design  for  the  external  load  alone. 

The  following  suggestions  may  serve  as  a  guide  in  practical 
prj')letiii  involving  the  resultant  load  on  bolts  when  the  initial 
load  due  to  screwing  up  is  apt  to  be  considerable  : 

(a)  If  the  bolt  is  manifestly  very  much  more  yielding  than  the 
connected  members,  design  the  bolt  simply  for  the  initial  load  or 
for  the  external  load,  which  ever  is  the  greater. 

(b)  If  the  probable  yield  of  the  bolt  is  from   one-half  to  once 
that  of  the  connected  members,  consider  the  resultant  load  as  the 
initial  load  plus  from  one-fourth  to  one-half  the  external  load. 

(*•)  If  the  yield  of  the  connected  members  is  probably  four  or 
five  times  that  of  the  bolts,  take  the  resultant  load  as  the  initial 
load  plus  about  three-fourths  the  external  load. 

(d)  In  case  of  extreme  relative  yielding  of  the  connected  mem- 
bers, the  resultant  load  may  be  assumed  at  nearly  the  sum  of  the 
initial  and  external  loads. 

63.  Proportions  and  Form's  of  Bolts  and  Nuts.      [Unwin, 
§§  86-88.] 

64.  Locking  Arrangements  for  Nuts.     [Uuwin,  §  89.] 

65.  Bolting  Plates.     [Unwin,  §§  90-91.] 

66.  Joint  Pins.     [Unwin  §  92.] 

The  sections  of  the  pin  should  be  checked  for  resistance  to 
shearing,  like  a  double  shear  rivet.  The  eyes,  or  bosses,  in  which 
the  pin  bears,  should  be  checked  against  tearing  out.  In  general  t 


the  strength  of  each  of  these  parts  should  at  least  equal  that  of 
the  rod  which  transmits  the  load  to  the  joint.  In  cases  where  the 
motion  at  the  joiut  is  considerable,  the  proportions  necessary  to 
secure  sufficient  area  of  the  bearing  to  avoid  undue  wear  may  give 
an  excess  of  strength.  The  proportions  of  btarings  and  journals 
will  be  treated  in  the  next  chapter. 

67.  Keys.     [Unwin  §§  93  to  98.] 

Sunk  keys  may  be  divided  into  three  classes  : — Flat  keys  ; 
square  keys  ;  and  feather  keys.  The  ordinary  form  of  key  for 
heavy  work  is  the  flat  key,  as  shown  by  Fig.  112  C  (Unwin). 
This  is  the  form  commonly  used  in  engine  work  and  construction 
of  mill  machinery,  etc.  As  it  is  tapered  top  and  bottom  and 
driven  in  hard  it  is  tightest  on  these  faces,  but  it  should  also  be 
fitted  on  the  parallel  sides.  While  this  key  resists  shearing,  it 
really  acts  as  a  diagonal  strut,  in  transmitting  force,  as  indicated 
in  Fig.  43  With  the  usual  proportions,  the  severest  action  on  the 
key  is  the  crushing  forces  at  the  bearing  surfaces  a,  a! .  The  re- 
sistance to  relative  rotation  of  the  connected  members  is  due  to 
the  resistance  of  the  key  to  crushing  and  shearing,  and  to  the 
friction  between  the  hub  and  shaft  at  b  (Fig.  43).  The  effect  of 
the  taper  of  the  key  is  to  somewhat  distort  the  hub  and  to  throw 
it  out  of  exact  concentricity  with  the  shaft  ;  this  latter  effect  being 
greater  because  the  hub  is  usually  bored  sufficiently  large  to  be 
easily  moved  to  place  along  the  shaft.  However,  with  the  usual 
dimensions  of  hub,  the  springing  of  the  work  is  not  apt  to  exceed 
limits  which  are  admissible  in  most  heavy  machines. 

The  tendency  of  the  hub  to  work  loose  on  the  shaft,  especially 
when  the  forces  acting  are  continually  reversed  in  direction,  can 
be  largely  overcome  by  placing  two  keys  "quartering",  as  shown 
in  Fig.  113  (Unwin). 

Unwin  states  that  a  taper  of  i*/2  inch  per  foot  corresponds  to 
about  the  angle  of  repose  for  oiled  steel  and  iron  surfaces.  Such 
a  steep  taper  of  the  key  would  not  give  a  good  grip  of  the  hub  on 
the  shaft,  and  the  shock  incident  to  operation  would  be  very  liable 
to  loosen  the  key.  The  usual  taper  of  keys  is  about  ^th  inch 
per  foot. 


Fig.  53. 


-113- 

If  the  thickness  (/),  or  depth,  of  the  ke)-  is  too  great,  the  shaft 
is  unduly  weakened  by  the  deep  key  seat.  In  practice,  the  fol- 
lowing proportions  are  usual  ones  :  b  =  %d;  /— ^2  b=}fad;  in 
which  d  —  the  diameter  of  the  shaft,  £  =  the  breadth,  and  /=the 
thickness  or  depth  of  the  key.  In  application,  the  dimensions 
would  be  to  the  nearest  TVth  inch.  The  above  proportions  are 
suitable  for  the  key  of  a  main  driving  pulley  or  gear  ;  that  is  for 
transmitting  a  twisting  moment  up  to  full  capacity  of  the  shaft. 

In  cases  where  light  pulleys  or  gears  are  carried  on  large  shafts, 
the  dimensions  of  the  keys  may  often  be  much  less  than  those  as- 
signed by  the  above  formulas.  The  form  of  key  above  discussed 
resists  axial  motion  of  the  hub  along  the  shaft  by  friction  between 
the  contact  surfaces. 

Where  the  power  is  distributed  through  pulleys  or  gears,  each 
carrying  only  a  small  portion  of  the  load,  set  screws  are  fre- 
quently used  as  a  substitute  for  keys.  These  are  convenient,  but 
of  doubtful  holding  capacity  under  any  but  quite  light  loads.  If 
they  slip,  the  shaft  is  apt  to  be  burred  up  so  that  it  is  difficult  to 
remove  the  pulley. 

If  the  twisting  moment  transmitted  through  the  key  is  7",  the 
pressure  against  the  bearing  faces  is  P=  T-r-r;  r  being  the  radius 
of  the  shaft.  The  area  of  each  bearing  face  is  ^  tl,  in  which  t  is 
thickness  of  the  key  and  /  is  the  length  of  hub.  The  crushing 
stress  per  unit  of  area  of  bearing  face  isy^  =  2  P-s-tl. 

In  members  of  machine  tools,  and  in  other  similar  cases,  the 
eccentricity  of  the  hub  which  is  liable  to  result  from  driving  a 
taper  flat  key  may  be  sufficient  to  appreciably  impair  the  accuracy 
of  operation.  Under  such  conditions,  particularly  if  the  service 
is  not  severe,  square  keys  may  be  advantageously  substituted  for 
flat  keys.  Fig.  44  illustrates  the  true  square  key.  It  should  be 
fitted  tightly  on  the  parallel  sides  a  a,  but  should  not  be  tight  at  top 
and  bottom.  It  is  evident  that  a  key  fitted  in  this  way  has  no 
tendency  to  throw  the  hub  out.  The  hub  should  be  bored  so  as 
to  fit  the  shaft  closely,  though  not  tight  enough  to  require  heavy 
driving  in  putting  the  hub  in  place. 

When  a  square  key  is  properly  fitted  the  stress  upon  it  from  the 
load  transmitted  may  be  taken  as  a  simple  shear  =/|  =  P-s-  bl ; 


in  which  b  is  the  breadth  and  /  the  length  of  the  key*  The  bear- 
ing faces  are  subjected  to  a  crushing  action,  similar  to  that  with  a 
flat  key.  In  this  type  of  keys,  b  is  usually  equal  to  /. 

Feather  keys  are  used  when  the  hub  occupies  different  posi- 
tions on  the  shaft ;  as  in  the  driving  gear  of  drill  press  spindles. 
In  such  cases  a  key  fixed  in  the  hub  slides  in  a  long  keyway,  or 
spline,  cut  in  the  shaft ;  or  a  long  key,  or  "  feather,"  is  fixed  in 
the  shaft,  and  the  keyway  of  the  hub  slides  along  this  "  feather." 
Of  course  the  fit  of  the  key  in  the  groove  along  which  it  slides 
must  be  a  "  sliding  fit. ' '  If  the  axial  motion  under  load  is  great, 
the  bearing  surface  of  the  sliding  elements  should  be  sufficient  to 
avoid  excessive  wear. 


VII. 

JOURNALS  AND  BEARINGS ;  THRUST  BEAR- 
INGS; GUIDES. 


68.   Outline  of  the  Functions  and  Operation  of  Bearings. 

[Unwin,  §§  104-105,  page  181]. 

The  leading  considerations  in  the  design  of  journals  and  bear- 
ings, or  other  guiding  elements  which  by  sliding  upon  each  other 
constrain  the  motion  between  two  members,  are  :  Form  ;  dimen- 
sions ;  materials  ;  lubrication. 

Where  the  two  members  are  required  to  have  relative  rotation* 
in  parallel  planes,  the  journal  and  bearing  must  have  contact  sur- 
faces which  are  a  pair  of  surfaces  of  revolution,  with  a  commons 
axis  coinciding  with  the  axis  of  relative  rotation.  The  most 
usual  form  for  such  constraining  elements  is  the  right  cylinder ; 
frequently  provided  with  collars  to  limit  the  end  play.  In  some 
cases  no  appreciable  end  motion  is  permissible,  and  in  few  cases 
would  this  play  be  more  than  a  small  fraction  of  an  inch.  The 
bearing  is  commonly  split  along  its  length  with  provision  for 
closing  it  on  the  journal  to  compensate  for  wear.  In  some  cases 
the  contact  surfaces  are  conical  with  a  means  of  moving  the  jour- 
nal axially  (relatively  to  the  bearing)  to  take  up  the  wear.  A 
method  of  taking  up  wear  in  conical  bearings  is  indicated  by 

FJR-  45- 

Two  members  not  moving  in  parallel  planes  are  sometimes  con- 
nected through  bearings  having  spherical  surfaces,  or  by  what  is 
commonly  called  a  "ball  and  socket  joint." 

If  the  desired  relative  motion  is  a  translation,  appropriate  guid- 
ing surfaces  (having  elements  parallel  to  the  line  of  motion)  are 
used. 

The  strength  of  a  journal  depends  upon  its  form,  dimensions, 
material,  and  method  of  support.  For  example,  an  overhang- 
ing crank-pin,  or  end  journal,  may  be  treated  as  a  cantilever 


—  H6  - 

of  circular  section,  in  which  the  working  strength  is  proportional 
to  the  cube  of  the  diameter,  inversely  as  the  length,  and  directly 
as  the  safe  stress  for  the  material.  A  journal  of  similar  form  and 
dimensions,  if  supported  at  both  ends,  is  evidently  much  stronger 
than  this  cantilever. 

When  a  journal  runs  in  its  bearing,  the  frictional  resistance  re- 
sults in  heat  and  in  wear  of  both  members.  In  some  cases,  as  in 
long  lines  of  transmission  shafting,  the  most  important  eifect  of 
friction  is  the  resulting  loss  of  energy.  In  other  cases,  as  in  en- 
gine crank-pins  and  eccentrics,  the  danger  of  heating  and  of 
resultant  injury  to  the  bearing  surfaces  is  paramount.  In  still 
different  classes  of  bearings,  as  those  of  grinding  lathes,  and  other 
light  machine  tools  for  producing  very  accurate  work,  the  most 
objectionable  result  of  friction  is  the  change  of  form  through  wear, 
which  affects  the  accuracy  of  constrainment  and  thus  impairs  the 
quality  of  the  product  of  the  machine.  The  danger  from  over- 
heating and  the  loss  of  energy  through  friction  are  usually  sec- 
ondary considerations  in  this  last  named  class  of  bearings. 

While  analysis  may  suggest  general  relations  between  lengths 
and  diameters  of  journals  applicable  to  these  special  classes,  the 
design  of  such  bearings  for  permanence  or  durability  must  be 
based  very  largely  upon  experience,  and  the  sizes  and  proportions 
in  general  use  are  the  result  of  a  process  of  evolution. 

Frictional  loss  is  mainly  dependent  upon  the  velocity  of  rub- 
bing at  the  journal  surface,  the  bearing  pressure,  and  the  lubrica- 
tion. With  perfect  lubrication  (w'hich  is  very  difficult  to  main- 
tain) the  friction  is  influenced  but  little,  if  at  all,  by  the  materials 
of  the  journal  and  bearing,  except  as  the  smoothness  of  the  sur- 
faces is  affected  by  the  nature  of  the  materials.  But  in  case  of 
failure  of  lubrication,  these  materials  have  an  important  influence 
on  the  consequences.  A  film  of  oil  between  the  journal  and  bear- 
ing completely  separates  their  surfaces  when  ideal  lubrication  ex- 
ists. No  surface  is  absolutely  smooth,  and  the  small  points  and 
irregularities  projecting  from  either  surface  tend  to  pierce  the  oil 
film  and  thus  to  abrade  the  opposite  surface.  The  smoother  the 
surface  the  less  will  be  this  tendency,  and  the  thinner  the  oil  film 


can  become  without  permitting  metallic  contact.  Aside  from  the 
susceptibility  to  smooth  finish,  the  nature  of  the  materials  com- 
posing the  journal  and  bearing  have  little  effect  on  the  action  as 
long  as  a  good  film  of  oil  is  maintained.  If,  however,  the  lubri- 
cation fails,  partially  or  completely,  and  the  condition  of  metallic 
contact  is  established,  one  or  both  of  the  members  may  be  seri- 
ously "  cut  "  before  the  machine  is  stopped  and  proper  conditions 
are  restored. 

As  stated  above,  the  friction  is  quite  independent  of  the  mate- 
rials of  the  journals  and  bearings  (aside  from  the  influence  of  fin- 
ish or  polish)  as  long  as  the  lubrication  is  properly  maintained. 
Different  bearing  materials  seem  to  be  capable  of  carrying  about 
the  same  intensity  of  pressure  with  good  lubrication,  provided  the 
maximum  intensity  of  pressure  is  below  that  at  which  the  mate- 
rial will  crush,  or  flow.  When  the  bearing  is  subject  to  consid- 
erable shock  or  "hammering,"  a  soft  metal  bearing  face  may 
have  the  oil  grooves  closed  up,  or  become  otherwise  so  deformed 
that  the  distribution  of  the  oil  will  be  impeded  and  lubrication  in- 
terrupted. It  happens,  not  infrequently,  that  bearings  run  hot 
from  such  cause.  Of  course  the  material  used  should  not  yield 
appreciably  tinder  any  compressive  action  to  which  it  is  apt  to  be 
subjected  in  service. 

It  appears  that  lead  and  zinc  alloys  are  more  liable  to  corrosion 
from  animal  or  vegetable  acids  than  are  tin  and  copper  ;  and  this 
probably  accounts  in  part  for  the  favor  in  which  genuine  Babbitt 
metal  (tin  88  per  cent,,  antimony  8  per  cent.,  copper  4  per  cent.) 
has  been  held.  Corrosion  roughens  the  surface,  and  may  cause 
serious  heating.  The  purely  mineral  oils  are  less  apt  to  act  upon 
the  material  in  this  way,  and  numerous  bearing  materials  con- 
taining lead  have  found  extensive  use  in  recent  practice. 

In  the  event  of  failure  of  lubrication,  the  metallic  surfaces  run 
in  actual  contact.  If  the  bearing  material  has  a  firmer  and 
stronger  structure  than  the  journal,  the  latter  is  most  abraded, 
and  the  particles  removed  from  it  tend  to  heap  up  at  one  point  on 
the  stationary  bearing,  cutting  the  journal  more  and  more  deeply 
at  each  turn.  If,  on  the  other  hand,  the  bearing  material  yields 


—  n8  — 

more  readily  than  that  of  the  journal  the  material  removed  ad- 
heres to  the  revolving  member  and  rotates  with  it,  thus  scoring 
the  surfaces  less  than  with  the  opposite  conditions.  Hence,  it  is 
generally  safer  to  use  a  bearing  metal  which  is  softer  than  the 
journal.  The  lining  of  a  bearing  is  generally  more  cheaply  re- 
placed than  the  journal,  and  it  is  therefore  desirable  to  have  such 
injury  as  occurs  confined  to  the  former,  as  far  as  it  can  be.  These 
reasons  indicate  why  it  is  not  usually  good  practice  to  make  jour- 
nal and  bearing  of  similar  materials.  There  are  exceptions  to  the 
rule  that  the  bearing  should  be  the  softer  material. 

In  cases  of  low  velocity  of  rubbing,  cast  iron  has  often  been  found 
to  give  good  results  when  running  on  cast  iron.  Cast  iron  piston 
rings  in  a  cast  iron  steam  cylinder  have  usually  been  found  to  fur- 
nish the  best  combination.  In  spindles  of  milling  machines  and 
grinding  machines  the  bearing  is  frequently  a.  hardened  steel 
bushing,  while  the  journal  is  of  softer  steel.  These  bearings  are 
often  conical  (see  Fig.  45),  and  the  wear,  if  concentrated  on  the 
rotating  shaft,  takes  place  quite  uniformly  all  around  ;  but  if  the 
wear  were  mainly  in  the  stationary  bearing  it  would  occur  on  the 
side  of  greatest  pressure.  It  will  readily  appear  that  the  compen- 
sation for  wear  affects  the  alignment  of  the  spindles  less  if  this 
wear  is  uniformly  distributed  around  the  journal. 

Bronze,  either  with  or  without  a  "white  metal"  lining,  is 
much  used  for  bearings.  The  chief  advantages  of  bronze  for  a 
bearing  surface,  proper,  are  its  resistance  to  compression  and 
shock,  and  less  liability  of  injuring  a  wrought  iron  or  steel  shaft 
than  an  iron  bearing  surface  When  bronze  is  used  in  a  bearing 
with  a  "Babbitt  "  or  other  lining,  the  bronze  is  not  the  true  bear- 
ing material,  unless  the  lining  metal  melts  and  runs  out.  The 
justification  for  use  of  bronze  rather  than  cast  iron  in  such  cases 
(as  in  connecting  rod  "  brasses  ")  is  in  its  superior  strength  under 
shock  to  cast  iron,  and  the  ease  with  which  it  is  fitted  up  ;  to- 
gether with  the  reduced  danger  to  the  shaft  in  case  the  bearing 
ever  becomes  so  hot  as  to  melt  out  the  "  Babbitt  "  before  the  ma- 
chine is  stopped. 


69.  General  Considerations  on  Friction.— [Unwin,  §§  106 
to  109].     If  the  journal  has  a  somewhat  smaller  diameter  than 
the  bearing,  as  in  Fig.  46,  the  contact  is  theoretically  (except  for 
the  influence  of  the  oil  film)  along  a  line  ;  and  the  load  on  the 
bearing  is  P,  equal  to  the  load  carried  by  the  journal.    Of  course, 
small  wear  of  the  bearing  would  result  in  distribution  of  the  load 
over  a  finite  area  on  the  bearing,  but  the  load  on   the  bearing 
would  remain  substantially  equal  to  P  with  any  moderate  amount 
of  wear.     If  the  journal  and  bearing  originally  have  the  same 
diameter,   as  indicated  by   Fig.  47,  the  bearing  pressure  would 
th  >o<-  tically  be  a  uniformly  distributed  normal  pressure  equal  to 

—  P.     A  bearing  could  not  run  long  under  this  condition,  for  the 

oil  film  could  not  be  maintained  with  such  a  fit.  The  condition 
indicated  by  Fig.  48  is  that  in  which  the  journal  is  initially  of 
larger  diameter  than  the  bearing.  In  this  last  case,  the  bearing 
pressure,  R  R,  might  be  infinity  except  for  the  yielding  of  the 
members  and  wear  ;  but  such  a  bearing  would  rapidly  change  to 
the  form  shown  in  Fig.  47,  and  would  tend  to  approach  that  of 
Fig.  46.  However,  the  surfaces  might  be  seriously  injured  dur- 
ing this  change,  and  the  successful  operation  of  a  bearing  ordina- 
rily requires  that  it  be  fitted  up  with  a  slightly  larger  diameter 
than  the  journal.  It  is,  therefore,  justifiable  to  treat  the  load  on 
the  bearing  as  equal  to  P,  and  the  friction  at  the  bearing  as  p.  P; 
p  being  a  special  coefficient  for  bearings,  as  explained  on  page 
183  of  Unwin  ;  though  the  oil  film  distributes  the  pressure. 

70.  Outline  of  the  Theory  of  Lubrication. — [Unwin,  §  no]. 
Oil  applied  to  the  surface  of  a  rotating  journal  tends  to  adhere  to 
this  surface  and  to  be  carried   around   with   the  journal.     The 
adhesion  of  the  oil  to  the  journal  is  the  means  of  transferring  it 
from  the  side  of  least  pressure,  where  it  should  be  introduced,  to 
the  loaded  side  of  the  bearing.     Of  course  the  oil,  upon  coming 
into  contact  with  the  stationary  surface  of  the  bearing,  adh«res  to 
this  surface  also  ;  so  that  a  layer  of  oil  next  the  journal  tends  to 
revolve,  and  a  layer  next  the  bearing  tends  to  remain  stationary. 
Owing  to  the  cohesive  action  between  the  particles  of  the  oil 


(viscosity)  a  resistance  is  offered  to  this  relative  motion  of  its  sep- 
rate  particles,  and  the  friction  of  a  well  lubricated  bearing  is  cine, 
mainly,  to  this  fluid  resistance.  The  globules  of  oil  tend  to  roll 
between  the  two  surfaces  like  balls  in  a  ball  bearing,  though  the 
action  is  more  like  that  which  might  be  imagined  to  exist  if  the 
balls  were  strongly  magnetized.  The  rotation  of  the  layer  of  oil 
next  to  the  journal  is  retarded  somewhat  on  account  of  the  ad- 
hesion of  the  oil  to  the  stationary  surface  of  the  bearing  and  by 
the  cohesion  of  the  intermediate  particles.  On  the  other  hand, 
the  adhesion  of  the  oil  to  the  journal  and  the  cohesion  between 
the  particles  tend  to  carry  the  oil  film  around  with  the  journal. 
Notwithstanding  the  pressure  on  the  loaded  side  of  the  bearing, 
some  of  the  oil  will  be  dragged  along  with  the' journal  into  the 
curved  wedge  shaped  space  between  the  journal  and  bearing  (see 
Fig.  46),  if  the  intensity  of  bearing  pressure  is  not  too  great  for 
the  viscosity  of  the  oil  used.  A  very  close  fitting  bearing,  as  in 
Fig.  47,  would  not  admit  the  oil  readily,  and  in  general  the 
edges  of  the  bearing  should  be  rounded  or  chamfered.  If  the  in- 
tensity of  bearing  pressure  is  low  enough  for  the  oil  used  (remem- 
bering that  the  viscosity  of  the  oil  becomes  reduced  as  the 
bearing  becomes  warm)  the  metal  surfaces  are  kept  separated  by 
the  film  of  oil.  As  shown  in  the  reports  of  Tower's  experiments, 
this  pressure  may  be  equivalent  to  300  or  400  pounds  per  square 
inch  of  projected  area  on  a  bearing  subjected  to  a  constant  load. 
If  the  pressure  on  the  bearing  is  intermittent,  the  intensity  of 
pressure  may  be  much  higher  than  these  figures.  In  the  case  of 
the  crank  pin  of  the  ordinary  steam  engine,  in  which  the  direc- 
tion of  pressure  completely  reverses  during  a  revolution,  pressures 
as  high  as  1,000  pounds  per  square  inch  of  bearing  are  frequently 
carried.  The  conditions  of  the  crank  pin  bearing  during  the 
opposite  strokes  of  the  piston  is  shown  by  Figs.  490  and  49^.  In 
the  former,  the  pressure  tends  to  force  the  oil  from  the  side  a  to  the 
side  £.;  but  the  reversal  of  pressure  on  the  return  stroke  tends  to 
return  the  oil  to  side  a.  The  sluggishness  of  the  flow  from  one 
side  to  the  other  prevents  the  complete  expulsion  of  the  oil  film 
from  the  pressure  side  during  the  short  time  ot  a  single  stroke  ; 


—   121    — 

while  a  similar  intensity  of  pressure  continuous  in  direction 
might  expel  the  oil.  The  conditions  are  even  more  favorable  at 
the  crosshead  pin,  in  horizontal  engines,  as  the  pressure  and  the 
direction  of  motion  between  the  journal  and  bearing  are  both 
reversed  at  each  stroke  ;  actions  which  tend  to  distribute  the  oil 
where  needed.  For  this  reason,  and  also  because  the  velocity  of 
rubbing  (hence  the  tendency  to  heat)  is  less  at  the  crosshead  pin, 
the  intensity  of  bearing  pressure  at  this  pin  is  usually  con- 
siderable greater  than  that  at  the  crank  pin.  These  two  pins 
carry  substantially  the  same  total  load,  but  the  intensity  of 
pressure  at  either  is  this  load  divided  by  the  projected  area  of  its 
bearing,  and  the  crosshead  pin  is  usually  considerably  smaller 
than  the  crank  pin.  In  vertical  engines,  the  difficulty  in  intro- 
ducing oil  at  the  top  of  the  crosshead  pin  results  in  less  favorable 
conditions  ;  because  the  motion  of  the  journal  is  not  sufficiently 
great  to  distribute  the  oil  over  the  top  bearing,  where  the 
pressure  is  applied  on  the  down  stroke. 

If  the  intensity  of  bearing  pressure  is  small,  a  light  bodied  oil 
can  be  used.  From  what  has  been  said  in  regard  to  the  friction 
being  mainly  due  to  the  fluid  resistance  of  the  oil  (with  thorough 
lubrication),  it  will  appear  that  a  thin,  fluid  oil  with  a  low  in- 
tensity of  pressure  is  generally  desirable,  on  the  score  of  reducing 
friction,  especially  for  high  speed  bearings.  However,  in  many 
cases  of  heavily  loaded  bearings  unduly  large  surfaces  might  be 
required  to  secured  a  low  intensity  of  pressure  upon  them. 
Furthermore,  if  the  reduction  of  pressure  is  obtained  by  increas- 
ing the  diameter  of  the  journal,  the  velocity  of  rubbing  is  corre- 
spondingly increased,  for  a  given  rotative  speed.  The  friction 
may  thus  be  decreased  by  the  larger  diameter  and  lower  pressure, 
which  permits  the  use  of  lighter  oil,  while  the  work  of  friction 
(the  product  of  the  friction  into  the  space  passed  over  against  this 
resistance)  would  not  be  decreased.  It  is  this  frictional  work 
which  measures  the  loss  of  energy.  If  the  bearing  pressure  is 
reduced  by  increase  of  the  length  of  the  journal,  the  velocity  of 
rubbing  is  not  increased,  but  a  limit  to  increase  of  bearing  area 
by  this  means  is  imposed  by  considerations  of  strength  and 
rigidity.  The  journal  is  often  a  beam  of  some  form,  the  strength 


and  rigidity  of  which  are  decreased  by  increasing  the  length 
unless  the  diameter  is  correspondingly  increased.  Even  if  the 
journal  is  not  in  danger  of  breaking  with  such  increase  of  its 
length  alone,  it  may  spring  under  its  load  enough  to  concentrate 
the  pressure  upon  a  small  portion  of  the  nominal  bearing  area 
and  thus  the  maximum  intensity  of  pressure  on  the  bearing  may 
be  as  great  as,  or  even  greater  than,  that  due  to  a  shorter  journal. 
See  Fig.  50,  which  indicates  this  action  to  an  exagerated  degree. 
A  journal  supported  on  both  sides  of  the  bearing  (Fig.  51)  can 
evidently  have  a  greater  length,  relative  to  its  diameter,  than  an 
overhung  journal  (Fig.  50),  other  things  being  equal.  If  the 
bearing  is  so  mounted  that  it  can  swivel  freely,  and  thus  accom- 
modate itself  somewhat  to  the  deflection  of  the  journal  (or  shaft), 
as  indicated  in  Fig.  52,  the  length  can  be  greater  relatively  to  the 
diameter  than  with  a  rigid  bearing.  Overhung  crank  pins  of  en- 
gines usually  have  lengths  not  exceeding  i^  diameters  :  the  main 
journals  of  engine  shafts  frequent!}'  have  lengths  equal  to  2 
diameters  ;  and  with  the  swivel  bearings  of  ordinary  lines  of 
shafting  the  length  of  bearing  is  quite  commonly  4  diameters. 
These  proportions  will  be  treated  in  a  later  section. 

71.  Point  of  Introduction  of  Oil. — [Unwin,  §  no.]  If  the 
oil  is  applied  nearly  opposite  the  point  of  maximum  pressure 
it  finds  ready  admission  ;  but  the  lubrication  cannot  be  satis- 
factorily accomplished  if  the  oil  is  introduced  at,  or  near,  the 
point  of  maximum  load,  unless  it  be  forced  in  by  a  pressure  ex- 
ceeding in  intensity  the  bearing  pressure  at  the  place  of  admis- 
sion. Mr.  John  Dewrance,  in  a  most  valuable  paper  before  the 
Institution  of  Civil  Kngineers  (Great  Britain)  on  "  Machinery 
Bearings,"  gives  the  following  rule:  "The  oil  should  be  intro- 
duced into  the  bearing  at  the  point  that  has  to  support  the  least 
load,  and  an  escape  should  not  be  provided  for  it  at  the  part  that 
has  to  bear  the  greatest  load."  If  this  rule  were  always  followed 
in.  construction,  the  elaborate  system  of  oil  channels  seen  in 
bearings  would  frequently  be  useless,  or  worse  than  useless. 
Direct  channels  from  the  oil  hole  to  near  the  ends  of  the  bear- 
ing, to  distribute  the  oil  latterly,  would  be  sufficient  for  most 
cases,  with  the  oil  hole  properly  placed.  The  results  quoted  in 


-J23  — 

Umvin,  page  188,  also  show  the  importance  of  adhering  to  this 
rule.  Many  instances  have  been  observed  of  the  oil  bubbling  up 
into  the  oil  cup,  when  this  simple  rule  has  been  neglected. 

72.  Theory  of  Journal  Proportions. — [Unwin,  §§  112  to  114.] 
As  pointed  out  by  Professor  Unwin  in  §  112,  the  proportions  of 
journals  as  found  in  practice  seem  to  agree  better  with  the  old 
assumption  that  p=a  constant  (the  law  of  dry  friction)  than  with 
those  laws  which  would  be  derived  from  Mr.  Tower's  experi- 
ments. The  dimensions  of  practice  are  generally  those  found  by 
experience  to  be  desirable.  For  everyday  running,  the  'propor- 
tions must  be  adequate  to  the  most  unfavorable  conditions  which 
arise  ;  not  simply  for  the  almost  perfect  lubrication  which  may 
be  maintained  in  the  laboratory.  The  common  oil  cup  feed  is,  at 
its  best,  less  effective  than  an  oil  bath,  and  it  may  fail  entirely. 
At  such  a  time  the  conditions  of  dry  friction  are  reached  if  the 
derangement  is  not  promptly  detected  and  remedied. 

In  equation  (3)  [Unwin, §  112],  the  quantity  ^P  is  the  friction 
{in  pounds)  and  it  d N  is  the  velocity  of  rubbing  (in  inches  per 
minute).  Hence,  the  work  of  friction  per  minute  in  ft.  Ibs.  = 

JJL  P  *--        .     This  must  be  divided  by  J  to  express  this  energy  in 

B.T.U.  per  minute.  The  surface  through  which  this  heat  is  dis- 
sipated is  TT  dl,  or  it  is  proportional  to  the  projected  area  =  dl, 
and  it  is  more  convenient  in  calculations  to  use  the  projected 
area.. 

An  increase  of  d  increases  the  surface  for  dissipation  of  the 
lieat ;  but  it  increases  the  velocity  of  rubbing,  hence  the  heat  to 
be  dissipated,  in  the  same  proportion.  It  appears  from  eq.  (4) 
[Unwin,  §  112]  that  the  heat  developed  per  square  inch  of  heat 
liberating  area,  and  consequently  the  temperature  attained,  is  in- 
dependent of  the  diameter  but  inversely  as  the  length  of  the  bear- 
ing. It  is  also  evident  that  the  wear  should  be  proportional  to 
the  friction  times  the  velocity  of  rubbing,  but  this  velocity,  and 
the  surface  over  which  the  wear  is  distributed,  both  vary  as  the 
diameter  ;  hence  the  wear  should  be  independent  of  the  diameter. 
On  the  other  hand,  eq.  (3)  shows  that  the  total  heat  developed, 
and  therefore  the  actual  energy  lost,  is  directly  proportional  to  the 


-  124  — 

diameter  and  independent  of  the  length.  It  thus  appears  desira- 
ble to  have  as  long  a  bearing  as  possible  to  secure  cool  running, 
with  as  small  a  diameter  as  possible  to  reduce  lost  work.  These 
conclusions  are  subject  to  the  following  "limitations,  however: 
(a)  If  the  diameter  is  too  small,  relatively  to  the  length,  the  deflec- 
tion of  the  journal  may  concentrate  the  pressure  unduly  (as  indi- 
cated in  Fig.  50)  even  if  there  is  no  danger  of  actual  breaking  of 
the  journal,  (b)  If  the  product  of  the  diameter  times  the  length 
is  too  small  the  intensity  of  bearing  pressure  may  expel  the 
lubricant.  The  considerations  involved  in  the  design  of  journals 
and  bearings  are,  therefore  :  I,  Heating  effect  and  wear.  II,  In- 
tensity of  bearing  pressure.  Ill,  Strength  or  rigidity.  IV, 
Energy  lost  through  friction. 

The  rational  procedure  in  determining  the  dimensions  of  a 
journal  and  bearing  would  seem  to  be  about  as  follows  : 

(a)  Determination  of  the  necessary  length  from  eq.  (4),  or  eq. 
(5),  Unvvin,  §  112 

(b)  Determination    of    the    diameter    necessary    to    give   the 
proper  intensity  of  bearing  pressure  with  the  length  found  in  (a). 

(c)  Check  of  the  journal   with    these   values   of  d   and  /  for 
strength  or  rigidity 

In  these  formulas  for  /,  the  value  of  P  is  to  be  taken  as  the 
mean  load  on  the  journal,  as  the  heating  is  an  effect  of  a  continu- 
ous action.  On  the  other  hand,  in  checking  for  strength  the 
maximum  load  must  obviously  be  used.  In  the  journals  of  en- 
gines and  many  other  machines  the  maximum  load  is  very  often 
much  in  excess  of  the  mean  load.  In  line  shafting,  these  two 
are  practically  equal  in  most  cases. 

.  The  values  of  intensities  of  bearing  pressures,  such  as  are 
given  by  Unwin  on  page  198,  should  be  considered  as  maximum 
values,  i.  e.,  as  about  the  maximum  allowable  values  obtained  in 
dividing  the  load  by  the  product  dl. 

The  proper  formula  for  checking  the  strength  or  rigidity  of  a 
journal  depends  upon  the  bending  moment,  or  other  straining 
action  on  the  journal  ;  thus,  for  the  strength  of  an  overhung  crank 

pin,  like  Fig.  137  (Unwin),  the  stress  =/=—  ^-.       In  the  gen- 


-  125  - 

eral  case  of  a  journal  subjected  to  a  bending  action  alone,  the 
stress  is  /=  ^2  — -  ;  in  which  M  =  the  bending  moment.  If 

7T     d 

there  is  a  combined  bending  and  twisting  action,  the  value  of  M 
in  this  expression  should  be  replaced  by  that  of  the  equivalent 
bending  moment,  (see  Art.  16 — Notes). 

Unfortunately,  the  design  of  journals  by  the  process  just  out- 
lined is  not  wholly  satisfactory  ;  because  the  proper  values  of  the 
constants  ft,  or  y,  of  eq.  (4),  or  (5),  [Unwin,  §  112],  are  not  defi- 
nitely determined  for  practical  cases.  These  constants  vary  so 
widely  with  apparently  small  changes  of  the  conditions  which 
govern  lubrication  that  it  is  not  safe  to  infer  their  values 
except  for  cases  very  similar  to  those  upon  which  observations 
have  been  made.  An  examination  of  several  Corliss  engines, 
each  of  which  had  a  value  of  (H.  P.)  H-  R=  13,  showed  values 
of  y  ranging  from  .32  to  .81  ;  or  crank-pin  lengths  ranging  from 
4.25"  to  10.5".  The  tables  of  Unwin,  on  page  193,  give,  for  rape 
oil,  ft  --  916,000  with  oil  bath  lubrication,  while  ft  =  310,000  for 
"siphon"  feed.  Professor  Unwin  says  (§114),  these  values 
"must  be  divided  by  a  factor  of  safety."  On  the  other  hand, 
higher  values  than  those  obtained  by  Mr.  Tower  (with  his  effect- 
ive oil  bath)  are  met  with  in  practical  operation  when  the  pres- 
sure is  intermittent,  notwithstanding  the  inferior  methods  of  lubri- 
cation in  these  latter  cases. 

While  this  theoretical  discussion  of  the  length  of  journals  is  im- 
portant in  giving  a  clear  conception  of  the  general  effects  of 
changes  in  relative  proportions  of  length  to  diameter,  it  does  not 
seem  to  afford  an  adequate  basis  for  assigning  actual  dimensions 
for  given  cases.  The  design  of  a  journal  must,  at  best,  depend 
largely  upon  judgment  ;  first,  because  of  the  uncertainty  and  del- 
icacy of  the  element  of  lubrication  ;  second,  because  the  journal 
must  be  given  such  dimensions 'as  will  insure  not  only  satisfactory 
running  under  usual  service  conditions,  but  a  fair  degree  of  insu- 
rance against  the  contingencies  that  will  probably  arise  during 
the  life  of  the  machine. 

The  intensity  of  bearing  pressure  (/>)  which  can  be  allowed  on 
a  given  class  of  journals,  with  velocities  of  rubbing  and  the  char- 


—  126  — 

acter  of  lubrication  usual  in  such  class,  is  more  definitely  deter- 
mined than  are  the  coefficients  ft  and  y.  The  allowable  working 
stress  for  the  common  materials  under  known  conditions  may  also 
be  quite  definitely  assigned.  These  quantities,  together  with  ratios 
of  length  to  diameter  which  have  been  found  generally  satisfactory 
in  practice,  probably  afford  the  most  reliable  data  for  design  of 
journals. 

In  journals  under  very  heavy  load  and  running  at  very  slow 
speed,  strength  is  the  primary  consideration,  intensity  of  bearing 
pressure  conies  second,  and  heating  may  need  but  little  attention. 
In  crank-pins  of  punching  machines,  for  example,  a  short  pin  of 
large  diameter  gives  the  best  combination  of  strength  and  bearing 
area.  At  higher  speeds  the  length  should  be  relatively  greater  ; 
and  in  very  high  speed  journals  under  small  straining  action 
much  greater  relative  lengths  are  appropriate.  These  points  can 
perhaps  be  best  developed  in  the  following  articles,  which  treat 
some  of  the  more  common  classes  of  jouinals  separately. 

73.  Main  Bearings  and  Crank-Pins  of  Punching  and 
Shearing  Machines.  —  In  these  machines,  and  others  of  a  simi- 
lar class,  the  rotative  speed  is  usually  low.  Though  the  load  on 
the  crank-pin  and  main  bearing  of  the  shaft  is  great  at  its  maxi- 
mum, this  load  is  only  applied  for  a  fraction  of  a  revolution. 
These  conditions  are  favorable  to  a  high  intensity  of  bearing  pres- 
sure. The  danger  from  over-heating  is  slight,  hence  the  length 
of  the  bearing  can  be  relatively  small.'  If  the  maximum  load,  P, 
is  assumed  to  be  uniformly  distributed  along  the  crank  pin,  the 
bending  moment  at  the  inner  end  of  the  pin  (where  the  moment 
is  greatest,  with  an  overhung  pin)  is  £  PI.  (See  Fig.  53). 

The  moment  of  resistance  of  the  pin  is  —  d*f. 


The  intensity  of  bearing  pressure  is 

p  p 

p  =  P-     .-.     «=£  (a) 


—    127  — 

From  eqs.  (i)  and  (2)  : 


Having  found  d,  its  value  can  be  substituted  in  the  following  ex- 
pression to  determine  /  : 


Should  the  value  of  /  be  greater  than  seems  desirable,  it  can  be 
reduced  by  increasing  d  to  maintain  the  required  bearing  area. 
Such  a  change  will  evidently  reduce  the  stress  in  the  pin. 

In  the  table  of  "Pressures  on  Bearings  and  Slides"  [Unwin, 
page  198],  the  allowable  value  of  p  is  given  at  3000  Ibs.  per  sq. 
inch  of  projected  area  for  such  bearings  as  those  treated  in  this 
article. 

Example  : — If  />=  70,000  Ibs.  ;  p  ==•  3,000  ;  f=  9,000. 

</=4  170,000=5.5" 

\  9,000  X  3,000     ^j 

Therefore,  /  =    -  T.0.-0*". .?._   =  4. 24" 

3,000X5-5 

To  design  the  main  journal  of  the  shaft,  the  bending  action  and 
the  load  at  this  bearing  should  be  known.  The  distance  between 
the  bearings  and  the  "overhang"  of  the  pin  must  be  known  to 
determine  these  (see  Fig.  53).  These  dimensions  would  be  fixed 
by  the  length  of  the  pin  and  the  design  of  the  frame,  and  are 
readily  determined  for  any  given  case. 

The  values  of  the  reactions  Rl  and  J?2,  and  the  bending  moment 
at  any  section  of  the  shaft,  can  readily  be  found  by  either  mathe- 
matical or  graphical  means,  when  /'and  the  above  mentioned  di- 
mensions are  known.  If  it  is  assumed  that  the  reaction  ^?,  acts 
at  the  middle  of  the  main  bearing,  the  bending  moment  is  a  max- 
imum at  the  point  where  A",  acts,  and  this  moment  is  M  =  Pa. 
The  twisting  moment  on  the  shaft  is  T=  Pr,  when  r  is  the  throw 


-    128  — 

of  the  crank,  or  one-half  the  stroke  of  the  machine.  The  bend- 
ing moment  equivalent  to  the  combined  bending  and  twisting 
actions  is 

M.=  $M+±SM*  +  T2  (5) 

The  assumption  of  M=Pa  is  rather  extreme,  as  the  bearing 
affords  a  quite  rigid  support  nearer  the  crank-pin  than  the  middle 
of  the  bearing.  The  reaction  and  the  moment  at  the  back  end  of 
the  shaft  is  much  less  than  that  at  the  main  bearing,  hence  the 
shaft  may  be  much  smaller  at  the  back  end. 

Equating  MK  to  the  moment  of  resistance  of  the  shaft,  and  solv- 
ing for  d : 


3    \ 


Example  : — If/*—-  70,000,  p=  3,000, /=  9,000,  as  in  the  ex- 
ample of  the  crank-pin  ;  and  a  =  6",  p  =  30",  r  =  1.5'', 

^=  70,000X6 .=  I4)00olbs. 
30 

R^  =  P  -f  RI  =--  84,000  Ibs. 

T=  70,000  X  1.5  =  105,000  inch  Ibs. 
M=  70,000  X  6  —  420,000  inch  Ibs. 
.'.  Me  =  426,500. 


d—  3  fro. 2  X  426,000  __  -  gc" 
\    9,000 

The  intensity  of  pressure  on  this  bearing  would  usually  be 
much  less  than  that  on  the  pin  ;  for.  as  illustrated  in  the  above 
examples,  the  diameter  of  the  main  journal  is  increased  over  that 
of  the  crank-pin  much  more  than  the  total  load  on  this  journal  is 
increased  over  that  of  the  pin,  and  the  main  journal  would  also  be 
the  longer.  In  this  example,  a  length  of  main  journal  of  only  3! 
inches  would  give  an  intensity  of  bearing  pressure  of  3,000  Ibs.  per 
sq.  inch  of  projected  aiea  ;  but  such  a  length  would  be  absurdly 
small.  If  the  length  of  this  journal  is  made  7  inches,  the  intensity 
of  bearing  pressure  would  be  less  than  1,600  Ibs.  per  square  inch. 


T 


Fig.  54 


!•— b- 


=E 


Fig.  55. 


Fig.  57 


Fig.  61. 


—  I29  — 

74.  Engine  Crank-Pins  ;  Side  Crank.  [Unwin,  §§  115,  116, 
118,  119,  120,  121.]  The  loads  on  engine  crank-pins  are  consid- 
erable, though  not  usually  so  excessive  as  in  punching  and  shear- 
ing machines  ;  while  the  velocity  of  rubbing  is  so  high  that  lia- 
bility to  heating  cannot  be  neglected. 

The  length  of  the  pin  is  a  more  important  consideration  than  in 
the  classes  of  journals  treated  in  the  preceding  article.  This 
length  could  first  be  determined  by  eq.  (4)  or  eq.  (5).  [Unwin, 
§  112],  except  for  the  uncertainty  as  to  proper  values  of  ft  or  y,  as 
referred  to  before. 

A  few  years  ago  the  principal  dimensions  of  a  large  number  of 
standard  commercial  engines  were  collected  for  use  in  Sibley  Col- 
lege. Upon  working  this  data  up,  the  engines  with  overhanging 
cranks  were  separated  from  those  with  centre  cranks.  The  di- 
mensions of  practice,  as  represented  by  about  seventy-five  side 
crank  engines  (mostly  of  the  Corliss  type),  are  expressed  more 
closely  by  the  formula  : 


than  by  the  theoretical  formula, 
,         H.P. 

/  =  y^T 

as  developed  by  Unwin  [eq    (5),  §  112]. 

In  the  above  equation  (i),  K  has  a  value  of  about  2",  so  that 
the  minimum  length  by  this  formula  would  be  more  than  2". 
The  value  of  c  ranges  from  0.2  to  0.4  ;  the  mean  value  being 
about  0.3.  It  may  then  be  said  that,  for  engines  of  this  class,  viz  : 
those  with  rotative  speeds  not  over  about  130  r.  p.  m.  and  from 
50  to  600  H.  P.  capacity,  the  average  length  of  crank-pin  is  about 

2".  (2) 


However,  the  value  of  c  varies  so  greatly  in  the  practice  observed 
that  it  is  not  a  satisfactory  basis  for  design,   and  the  following 
method  seems  preferable. 
9 


-  130- 

With  side  crank  engines  (overhanging  pins)  the  ratio  of  the 
length  to  the  diameter  of  the  pins  (A)  is  usually  from  i  to  i^  ; 
the  length  being  properly  somewhat  greater,  relatively,  at  the 
higher  rotative  speeds.  Unwin  gives  the  relation, 

A  =  -  = 

as  desirable  for  journals  in  general  ;  but  this  rule  cannot  be  ap- 
plied to  overhung  crank  pins.  With  a  uniformly  distributed  load 
on  an  overhung  crank  pin, 


But  if  p  is  the  intensity  of  bearing  pressure  per  square  inch, 

P(dr>=pt 

.-.    p*'l=P*.:   ^^  =  fx.  (4) 

Equating  (3)  and  (4)  and  solving  for  A  : 


From  eq.  (5)  the  value  of  A  corresponding  to  any  assigned  values 
of/  and  p  can  be  found.  If  A  is  taken  at  a  lower  value  than  that 
given  by  this  last  formula,  the  intensity  of  stress  will  be  less  than 
that  assumed,  and  the  pin  will  presumably  be  safe.  If/=  9000 
and  p  =  looo,  A  =  1^3.  It  thus  appears  that  the  limits  of  A  as 
given  above  (i  to  i^)  are  safe  for  wrought  iron  or  steel  pins  with 
about  the  customary  maximum  intensity  of  crank-pin  bearing 
pressure. 

From  the  relations  of  eq.  (5)  it  is  found  that/=  5.  i  p  A2.       (6) 
Substituting  5.1  /A2  for/,  in  eq.  (3),  and  solving  for  d, 


The  dimensions  of  the  crank-pin  may  then  be  found  as  follows  : 

(a)  Find  the  value  of  X,  from  eq.  (5),  corresponding  to  proper 
values  off  and  p. 

(b)  With  a  value  of  X  not  exceeding  that  determined  as  in  (a), 
find  d  from  eq.  (7). 

(c)  Determine  /from  the  relation, 

L=  A,     .-.     l  =  \d.  (8) 

d 

Example  :  An  engine  16"  X  20",  runs  under  a  boiler  pressure 
of  1  20  Ibs.  per  sq,  inch  (gauge  pressure).  If  f  can  be  8,000  Ibs. 
per  sq,  inch,  and/>  =  1,000,  determine  the  proper  dimensions  for 
the  overhung  crank-pin.  Assuming  that  the  engine  may  be  run 
condensing,  the  maximum  unbalanced  pressure  on  the  piston  may 
be  about  132  Ibs.  per  sq.  inch..  This  corresponds  to  a  total  load 
(P)  on  the  piston  of  about  26,500  Ibs.;  say  27,000  Ibs. 


=     I 
\ 


1 5  i  X  looo 
d= 


8-000       =1.25.     Taking  X  =  ij^,  from  eq.  (7) 


\iooo  X  1.125 

/=  9    X  4.9  =  5.52.      The  diameter  may  be  taken  at  5",  and 
8 

the  length  at  5^". 

75.  Crank  Shaft  Journals.  Side  Cranks.  [Unwin,  §  126.] 
The  strength  of  shafts  will  be  treated  more  fully  in  the  next 
chapter,  but  the  method  of  determining  the  journal  dimensions 
will  be  outlined  in  this  article.  To  determine  the  load  at  the 
main  journals,  the  resultant  reaction,  R^  (Fig.  54)  must  be 
known.  When  the  distances  a,  b,  c,  and  the  direction  .and 
magnitude  of  the  load  on  the  crank  pin,  the  total  belt  pull, 
and  weight  of  fly  wheel  are  known,  complete  data  is  avail- 
able for  computing  the  straining  actions  on  the  shaft  and 
the  dimensions  of  the  journals.  The  distance  a  will  often 
be  about  4  to  4^  times  the  diameter  of  the  crank  pin  ;  and 


-132- 

it  may  be  so  assumed  for  preliminary  computations,  in  ab- 
sence of  more  definite  data.  The  bending  moment  at  the 
centre  of  the  main  journal,  Fig.  54,  (assuming  the  load  as 
uniformly  distributed  along  its  length)  is  M=  Pa  ;  the  twisting 
moment  is  T=PR;  and  the  equivalent  bending  moment  can 
be  found  as  in  art.  73,  by  eq.  (5).  The  diameter  of  the  shaft  can 
then  be  found  by  eq.  (6)  of  art.  73.  The  length  (/)  of  the  shaft 
main  journal  is  often  about  twice  its  diameter  but  the  projected 
area  (/^)  should  be  sufficient  to  keep  the  intensity  of  bearing 
pressure  (/>)  within  safe  limits.  TJjiwin  gives  p  at  from  150  to 
to  250  Ibs.  per  sq.  inch.  The  total  load  on  the  main  journal  is 
^i  (Fig-  54/.  which  is  considerably  greater  than  the  load  on  the 
crank  pin  due  to  the  steam  pressure  alone.  When  the  total  re- 
action Rx  cannot  be  determined,  the  projected  area  of  the  main 
journal  may  be  taken,  for  ordinary  power  engines,  as  about  the 
maximum  load  on  the  piston  divided  by  175.  This  intensity  of 
pressure  is  much  less  than  that  allowed  for  the  crank  pin,  because 
the  component  of  the  load  on  the  shaft  bearings  due  to  fiy  wheel 
weight  and  belt  pull  is  not  intermittent  like  that  due  to  the  steam 
pressure. 

If  the  required  projected  area  makes  the  length  of  bearing  such 
that  the  distance  a  is  much  greater  than  that  assumed,  it  may  be 
necessary  to  revise  the  computations. 

The  length  of  the  crank  hub  (marked  B  in  Fig.  54)  is  usually 
somewhat  less  than  the  diameter  of  the  shaft. 

The  "outboard"  bearing  carries  a  much  smaller  load  (/?.,)  than 
the  main  bearing  ;  and  when  the  power  is  delivered  by  the  fly 
wheel,  that  portion  of  the  shaft  between  the  wheel  and  this  out- 
board bearing  is  not  subjected  to  torsion.  Hence  the  latter 
bearing  may  be  much  smaller  than  the  main  bearing. 

76.  Centre  Crank  Shafts  and  Pins.  Centre  cranks,  which 
have  main  bearings  each  side  of  the  crank,  are  used  on  many  high 
speed  engines,  on  marine  engines,  and  in  other  cases.  In  single 
crank  stationary  engines  with  this  type  of  shaft  there  are  fre- 
quently two  fly  wheels,  one  at  each  end  of  the  shaft.  Sometimes 
there  are  belts  on  both  wheels,  when  part  of  the  power  is  given  off 


-133- 

by  each  wheel  ;  sometimes  one  wheel  delivers  all  of  the  power ; 
and  in  other  cases  the  power  is  delivered  through  direct  connec- 
tion with  one  or  both  ends  of  the  shaft,  the  wheels  acting  purely 
as  fly  wheels.  The  weights  of  the  wheels,  the  total  belt  pulls, 
the  pressure  on  the  piston,  and  the  distances  a,  b,  etc.,  must  be 
known  to  completely  determine  the  straining  action  at  any  sec- 
tion of  the  shaft  and  the  load  on  any  bearing.  When  the  full 
data  is  at  hand,  the  combined  bending  and  twisting  actions  at  the 
critical  sections  can  be  found,  and  the  necessary  diameters  of 
such  sections  for  strength  can  then  be  computed  by  methods 
similar  t.o  those  of  preceding  sections  ;  that  is  by  placing 

Me  =    ff-  d*f,  and  solving  for  d. 

The  crank  pin  for  a  shaft  such  as  that  of  Fig.  55  must  be  much 
larger  than  an  overhung  pin  under  a  similar  Toad  ;  for  this  centre 
crank  pin  is  really  a  portion  of  a  beam  supported  at  R\  and  J?3 
and  loaded  at  P,  W}  and  W^  The  stresses  are  frequently 
severe  at  the  junctions  of  the  phi  and  of  the  two  outside  portions 
of  the  shaft  with  the  crank  arms,  and  liberal  "  fillets  "  should  be 
provided  at  these  angles.  The  analysis  of  the  straining  actions 
on  such  a  shaft  will  be  treated  more  fully  in  the  next  chapter. 

It  is  quite  common  practice  to  give  the  pin  of  an  engine  which 
has  a  centre  crank  a  diameter  equal  to  that  of  the  shaft  at  the  main 
bearings.  S  >me  engine  builders  make  the  pin  even  larger  than 
the  shaft,  while  a  few  of  them  make  it  somewhat  smaller.  For 
the  ordinary  case  it  will  be  well  to  make  the  diameter  of  the  pin 
as  large  as  that  of  the  main  journals. 

In  the  absence  of  any  more  definite  data,  the  following  method 
may  be  used  for  approximating  the  dimensions  of  the  crank  pin 
and  main  journals.  L,et />—  -the  maximum  load  on  the  crank 
pin,  due  to  the  steam  pressure  on  the  piston  ;  H.P.  =  the  horse 
power  of  the  engine  at  rated  load  ;  N~  the  revolutions  per  min.; 
d  =  the  diameter  of  the  shaft  =  diameter  of  the  crank  pin.  The 
diameter  of  the  shaft  should  be 


=  c   \H-P- 

L  \J~W~ 


(0 


-  134  - 

Unvvin  gives  (page  225)  the  value  of  C  in  eq.  (i)  as  4.55  for 
marine  engines.  For  stationary  high  speed  engines  up  to  250  or 
300  H.P.  C  is  usually  between  6.5  and  8.5  ;  the  general  average 
of  practice  corresponding  to  a  value  of  C=  7.3,  (about).  The 
approximate  diameter  may  be  found  by  eq.  (i),  using  a  proper 
value  of  C. 

When  the  diameter  of  crank  pin  has  been  fixed,  the  length  can 
be  found  from  the  allowable  intensity  of  bearing  pressure  (/>) 

pdl=P;    .-.1=—.  (2) 

pd 

The  average  value  of  p  may  be  taken  at  about  450  Ibs.  per  sq. 
inch.  This  is  less  than  half  the  intensity  of  pressure  frequently 
carried  with  overhung  pins.  The  difference  is  accounted  for  as 
follows:  The  length  of  pin  is  the  element  which  most  affects  the 
tendency  to  heat,  as  shown  by  Unwin,  §  112;  hence  this  length 
should  be  independent  of  the  diameter.  But  the  diameter  of  pin, 
necessary  for  strength,  is  much  greater  with  the  centre  crank 
type,  therefore  the  bearing  area  (dl)  would  be  correspondingly 
greater  with  a  proper  length,  or  the  intensity  of  pressure  would 
be  correspondingly  less  than  with  an  overhung  pin. 

The  intensity  of  bearing  pressure  on  the  main  journals,  due  to 
the  steam  pressure  on  the  piston  alone,  is  often  only  from  100  to 
125  Ibs,  per  sq.  inch.  Assuming  one-half  of  P  to  be  carried  by 
each  of  the  main  journals,  the  length  of  each  is  found  from  the 
following  : 

(3) 


In  which  d  has  the  value  determined  by  eq.  (i),  above. 

The  distance  from  centre  to  centre  of  main   bearings  (a  -f  a  - 
A)  of  Fig.  55)  may  be  roughly  determined  thus  : 


An  examination  of  a  considerable  number  of  standard  engines 
shows  that  the  value  of  Arranges  from  about  80  to  no,  and  a  fair 
average  value  for  use  in  preliminary  computations  is  about  90. 


-135- 

This  value  is  for  solid  forged  shafts.  If  the  shaft  is  "  built  up  " 
by  forcing  the  shaft  and  pin  into  crank  arms  or  discs,  the  span  A 
would  usually  be  greater.  The  assumption  that  the  shaft  is  a 
beam  supported  at  the  centre  of  the  bearings  is  in  the  nature  of 
an  error  on  the  safe  side,  as  the  effective  span  is  probably  consid- 
erably less,  owing  to  the  rigidity  of  the  bearings. 

The  preceding  method  is  only  to  be  considered  as  approximate, 
yet  it  will  perhaps  meet  the  requirements  in  many  cases.  When 
the  design  has  advanced  sufficiently  to  furnish  the  full  data,  the 
dimensions  assigned  by  the  foregoing  procedure  should  be 
checked. 

77.  Line  Shaft  Bearings  The  bearings  of  lines  of  shafting 
for  transmitting  and  distributing  power  are  generally  so  supported 
that  they  can  .swivel  to  a  certain  extent  to  accommodate  the  bear- 
ing surface  to  springing  of  the  shaft.  See  Fig.  52  and  Fig.  167 
(Unwin).  This  freedom  of  the  bearing  avoids,  to  a  very  consid- 
erable degree,  concentration  of  pressure  at  the  edge  of  the  bear- 
ing due  to  deflection  of  the  shaft,  such  as  is  indicated  by  Fig.  51, 
and  the  bearing  can  be  longer  than  would  be  practicable  with 
rigid  boxes  or  pillow-blocks.  It  is  quite  common  practice  to 
make  the  length  of  these  line  shaft  swivel  bearings  about  4  times 
the  diameter.  The  diameter  of  the  bearing  is  the  same  as  that  of 
the  shaft  between  bearings.  These  shafts  are  subjected  to  consid- 
erable bending,  as  well  as  to  the  torsion  due  to  the  power  trans- 
mitted, when  pulleys  or  gears  delivering  energy  are  at  some 
distance  from  a  bearing. 

If  the  shaft  is  subjected  to  torsion  only,  the  twisting  moment  is, 
as  given  in  art.  n, 

7^=63,024 — '— --  -^-d*f.  (i) 

In  which  H.  P.  •=.  the  horse  power  transmitted  and  N  the  revs, 
per  min. 

If  the  intensity  of  stress,  ft  is  taken  as  a  little  less  than  9,000 
Ibs.  per  sq.  inch,  the  diameter  of  the  shaft  is 

^=3-3' 


-136- 

When  the  bending  action  is  so  great  that  it  must  be  considered, 
the  equivalent  twisting  moment  Te  should  be  used  instead  of  T. 
See  art.  16,  eq.  (4).  In  such  cases  the  constant  of  eq.  (2)  is  too 
small. 

If  the  span  between  bearings  does  not  exceed  about  40  diame- 
ters, and  if  there  be  no  heavy  belt  pull  or  gear  thrust  far  from  a 
bearing,  the  diameter  may  be  taken  as 


The  constant  in  eq.  (3)  is  suitable  for  ordinary  shop  distribution 
shafting,  For  "head  shafts"  or  other  cases  where  the  bending 
action  is  severe,  it  will  be  well  to  make 


78  Crosshead  Pins  The  usual  type  of  short  crosshead  pin 
is  supported  at  both  ends.  The  diameter  necessary  for  securing 
the  bearing  area  (with  such  lengths  as  are  generally  convenient) 
gives  excess  of  strength  when  the  full  sized  pin  is  fitted  into  the 
crosshead.  If,  as  is  sometimes  the  case  in  the  familiar  "spade- 
handle  "  type  of  crosshead,  the  crosshead  pin  is  a  bushing  held 
in  place  by  a  smaller  pin  passing  through  it,  this  smaller  pin 
should  be  checked  for  resistance  to  shearing  ;  but  such  a  con- 
struction can  be  easily  avoided.  Occasionally  the  crosshead  pin 
is  of  such  form  that  the  bending  action  must  be  considered.  In 
this  case  it  should  be  checked  for  strength  as  a  beam 

The  pin  can  be  designed,  in  the  ordinary  case,  simply  with 
reference  to  the  allowable  intensity  of  bearing  pressure. 

P,  .'.  dl=P-*-p.  (i) 


The  length  is  commonly  from  i  to  1^2  times  the  diameter  ;  but 
as  the  velocity  of  rubbing  is  small,  length  of  pin  is  not  of  first 
importance,  and  this  ratio  can  be  varied  as  dictated  by  considera- 
tions of  the  general  design  of  the  crosshead. 


-  137  - 

7Q.  Bearings  of  Machine  Tools,  etc.  [Unwin,  §  117.] 
In  most  machine  tools,  and  in  many  other  machines,  the  loads 
upon  the  journals  are  comparatively  small,  and  are  too  indefinite 
to  become  the  basis  for  satisfactory  numerical  computations. 
The  requirements  of  rigidity  and  permanence  of  form  (minimum 
wear)  often  lead  to  the  adoption  of  journals  much  larger  than 
any  considerations  of  strength  would  dictate.  Liberal  bearing 
areas  are  the  rule  in  such  classes  of  machines.  As  the  speed  in- 
creases the  length  of  bearing  should  increase,  if  other  considera- 
tions permit.  A  general  guide  to  determination  of  length  may 
be  found  in  §  117  of  Unwin. 

80.  Thrust  Bearings.  [Unwin,  §§  130-131.]  The  intensity 
of  pressure  (/>)  on  a  thrust  bearing  may  be  computed  by  the  fol- 
lowing formula,  when  the  velocity  of  rubbing  (z/)  is  from  30  to 
170  feet  per  minute  at  the  outer  edge  of  the  step, 


This  expression  gives  an  intensity  of  750  Ibs.  at  a  velocity  of  30 
feet  per  minute,  and  of  50  Ibs.  at  170  feet  per  minute  :  which  may 
be  taken  as  about  the  limits  of  pressure  for  steps  or  thrust 
bearings. 

Kxcept  with  low  rubbing  velocities,  the  pressures  should  be 
quite  moderate;  because,  unlike  ordinary  journals,  the  wear  on 
different  portions  of  the  contact  surfaces  is  very  different.  The 
rubbing  velocity  is  zero  at  the  centre,  and  it  increases  with  the 
radius  to  a  m  iximum  at  the  outer  edge.  Hence  the  tendency  to 
wear  is  greater  at  the  outer  part.  Such  wear  concentrates  the 
pressure  near  the  centre,  on  a  smaller  area,  than  the  nominal 
bearing  surface.  The  effect  of  this  action  is  to  increase  the  real 
intensity  of  pressure  and  induce  cutting.  A  large  surface  and  a 
low  intensity  of  pressure  reduces  the  rate  of  wear  ;  but  a  more 
effective  means  of  securing  durability  is  co  use  the  collar-thrust 
bearing,  as  shown  in-  Fig.  139  (Unwin).  With  such  a  collar 
bearing,  the  difference  between  maximum  and  minimum  veloci- 
ties of  rubbing  depends  upon  the  ratio  of  the  outer  diameter  of 
collar  (dt}  to  the  inner  diameter  (d.^.  To  keep  this  difference  as 


small  as  possible  and  also  to  reduce  the  frictional  work  to  a  mini- 
mum by  avoiding  an  unnecessarily  high  velocity,,  the  outer 
diameter  should  not  be  very  much  larger  than  the  shaft  diameter. 
The  required  surface  can  be  supplied  by  increasing  the  number  of 
rings.  Of  course  practical  considerations  limit  the  increase  in 
number  and  the  decrease  in  size  of  the  collars. 

81.  Construction  of  Bearings,  Pedestals,  etc.  [Unwin, 
Chapter  VIII,  pages  244  to  268.]  The  almost  universal  practice 
in  this  country  is  to  use  the  swivel  form  of  bearing  for  line  shaft- 
ing, Fig.  167  (Unwin),  or  its  equivalent  This  is  used  not  only 
with  pillow  blocks,  as  shown,  but  with  post-hangers  (Unwin, 
Fig.  170),  and  drop-hangers  (Unwin,  Fig.  172).  The  bearing  is 
usually  of  cast  iron  without  any  lining  material,  as  this  metal  is 
satisfactory  under  the  low  intensity  of  bearing  pressure  possible 
with  these  long  swivel  bearings. 

For  rigid  bearings,  Babbitt  metal,  or  someotherso  called  ''anti 
friction"  alloy,  is  quite  commonly  used  as  a  lining.  For  many 
purposes  this  lining  is  simply  cast  in  the  box  around  the  shaft  or 
around  an  arbor.  If  cast  on  the  shaft  with  which  it  is  to  be  used, 
the  shaft  should  be  wrapped  with  paper,  which  gives  clearance  in 
the  bearing  and  also  reduces  the  danger  of  springing  the  shaft  by 
the  heat.  If  the  shaft  is  of  cold  rolled  steel,  the  danger  of  spring- 
ing it  is  considerable.  In  all  such  cases  it  is  preferable  to  cast  the 
lining  around  a  special  arbor  of  slightly  larger  diameter  than  that 
of  the  shaft  to  be  run  in  the  bearing. 

The  shrinkage  of  the  lining  metal,  in  cooling,  is  apt  to  leave  it 
somewhat  loose  in  the  supporting  shell.  For  the  classes  of  work 
requiring  greater  accuracy,  or  where  the  pressures  on  the  bearing 
are  apt  to  be  severe,  the  proper  procedure  is  to  cast  the  lining  on 
an  arbor  considerably  smaller  than  the  shaft  ;  then  to  compress 
or  "  pene  "  the  metal  after  it  has  cooled,  and  finally  bore  it  out 
to  the  required  size.  This  compression  of  the  soft  metal  causes 
it  to  fill  the  containing  cavity  snugly,  correcting  the  effect  of 
shrinkage;  it  also  gives  a  firmer  material  and  one  less  liable 
to  be  "  hammered  "  out  of  shape  under  service  load. 

Professor  Unwin  says  (page  250)  the  end  play  of  shafts  may  be 
one-tenth  the  length  of  journal.  This  is  an  extreme  amount, 


-  139  - 

and  the  collars  would  seldom  be  set  to  allow  as  much  end  play  in 
bearings  of  ordinary  dimensions. 

In  long  lines  of  shafting  the  expansion  or  contraction  with 
changes  of  temperature  may  be  considerable.  For  this  reason  the 
•collars  for  longitudinal  constrainment  should  be  placed  near  the 
point  at  which  it  is  desired  to  have  the  least  axial  motion.  Thus, 
if  there  should  be  a  bevel  gear  along  the  line,  the  collars  should 
be  placed  at  the  nearest  bearing  to  such  gear.  If  there  are  no 
gears,  however,  the  collars  may  be  placed  near  the  middle  of  the 
line,  or  at  any  point  where  there  is  special  reason  for  maintaining 
the  longitudinal  position  of  a  pulley  or  clutch. 

82.  Rectilinear  Sliding  Surfaces.  In  the  design  of  slides 
and  guides  for  securing  relative  translation  between  two  mem- 
bers, the  baaring  area  should  be  taken  with  reference  to  the  in- 
tensity of  bearing  pressure.  This  intensity  should  be  less  as  the 
velocity  of  rubbing  becomes  greater.  The  values  given  by 
Professor  Unwin  in  §  121  may  be  used  for  engine  crosshead 
shoes  ;  though  with  small  high  speed  engines  the  intensity  of 
bearing  pressure  is  often  not  more  than  20  to  30  Ibs.  per  sq.  inch. 

The  bearing  surfaces  of  the  slides  of  machine  tools  should  be 
liberal.  They  may  well  be  as  large  as  it  is  convenient  to  make 
th  MII  ;  for  comparatively  slight  wear  impairs  the  accuracy  of  the 
output  of  such  machines. 

The  following  discussion  of  slides  is  taken,  by  permission, 
from  the  work  on  Machine  Design  by  Professor  Albert  W.  Smith, 
of  Iceland  Stanford  University  : 

"  So  much  of  the  accuracy  of  action  of  machines  depends  on  the 
sliding  surfaces,  that  their  design  deserves  the  most  careful 
attention.  The  perfection  of  the  cross-'sectional  outline  of  the 
cylindrical  or  conical  forms  produced  in  lathes,  depends  on  the 
perfection  of  form  of  the  spindle.  But  the  perfection  of  the  out- 
lines of  a  section  through  the  axis  depends  on  the  accuracy  of  the 
sliding  surfaces.  All  of  the  surfaces  produced  by  planers,  and 
most  of  those  produced  by  milling  machines,  are  dependent  for 
accuracy  on  the  sliding  surfaces  in  the  machine. 

Suppose  that  the  short  block  A,  Fig.  56,  is  the  slider  of  the 
slider-crank  chain,  and  that  it  slides  on  a  lelatively  long  guide, 


D.  The  direction  of  rotation  of  the  crank,  A,  is  as  indicated  by 
the  arrow.  B  and  C  are  the  extreme  positions  of  the  slider.  The 
pressure  between  the  slider  and  the  guide  is  greatest  at  the  mid- 
position,  A,  and  at  the  extreme  positions,  B  and  C,  it  is  only  the 
pressure  due  to  the  weight  of  the  slider.  Also  the  velocity  is  a 
maximum  when  the  slider  is  in  its  mid-position,  and  decreases 
toward  the  ends,  becoming  zero  when  the  crank,  A-,  is  on  its 
center.  The  work  of  friction  is  therefore  greatest  at  the  middle, 
and  is  very  small  near  the  ends.  Therefore  the  wear  would  be 
greatest  at  the  middle,  and  the  guide  would  wear  concave.  If 
now  the  accuracy  of  a  machine's  working  depends  on  the  per- 
fection of  A'f>  rectilinear  motion,  that  accuracy  will  be  destroyed 
as  the  guide,  D,  wears.  Suppose  a  g  b,  E  FG,  to  be  attached  to 
A,  Fig.  57,  and  to  engage  with  D,  as  shown,  to  prevent  vertical 
looseness  between  A  and  D.  If  this  gib  be  taken  up  to  compen- 
sate wear  after  it  occurred,  it  would  be  loose  in  the  middle 
position  when  it  is  tight  at  the  ends,  because  of  the  unequal 
wear.  Suppose  that  A  and  D  are  made  of  equal  length,  as  in 
Fig.  58.  Then  when  A  is  in  the  mid-position  corresponding  to 
maximum  pressure,  velocity,  and  wear,  it  is  in  contact  with  D 
throughout  its  entire  surface,  and  the  wear  is  therefore  the  same 
in  all  parts  of  the  surface.  The  slider  retains  its  accuracy  of 
rectilinear  motion  regardless  of  the  amount  of  wear,  the  gib  may 
be  set  up,  and  will  be  equally  tight  at  all-  positions. 

If  A  and  B,  Fig.  59,  are  the  extreme  positions  of  a  slider,  D 
being  the  guide,  a  shoulder  would  be  finally  worn  at  C.  It 
would  be  better  to  cut  away  the  material  of  the  guide,  as  shown 
by  the  dotted  line.  Slides  should  always  "  wipe  over  "  the  ends 
of  the  guide  when  it  is  possible.  Sometimes  it  is  necessary  to 
vary  the  length  of  stroke  of  a  slider,  and  also  to  change  its  posi- 
tion relatively  to  the  guide.  Bxamples :  "Cutter  bars"  of 
slotting  and  shaping  machines.  In  some  of  these  positions, 
therefore,  there  will  be  a  tendency  to  wear  shoulders  in  the 
guide  and  also  in  the  cutter  bar  itself.  This  difficulty  is  overcome 
if  the  slide  and  guide  are  made  of  equal  length,  and  the  design  is 
such  that  when  it  is  necessary  to  change  the  position  of  the 


cutter  bar  that  is  attached  to  the  slide,  the  position  of  the  guide 
may  be  also  changed  so  that  the  relative  position  of  slide  and 
guide  remains  the  same.  The  slider  surface  will  then  just  com- 
pletely cover  the  surface  of  the  guide  in  the  mid-position,  and  the 
slider  will  wipe  over  each  end  of  the  guide,  whatever  the  length 
of  the  stroke. 

In  many  cases  it  is  impossible  to  make  the  slider  and  guide  of 
equal  length .  Thus  a  lathe  carriage  cannot  be  as  long  as  the  bed  ; 
a  planer  table  cannot  be  as  long  as  the  planer  bed,  nor  a  planer 
saddle  as  long  as  the  cross-head.  When  these  conditions  exist 
special  care  should  be  given  to  the  following  ;  I.  The  bearing 
surface  should  be  made  so  large  in  proportion  to  the  pressure  to  be 
sustained  that  the  maintenance  of  lubrication  shall  be  insured 
under  all  conditions.  II.  The  parts  which  carry  the  wearing 
surfaces  should  be  made  so  rigid  that  there  shall  be  no  possibility 
of  the  localization  of  pressure  from  yielding. 

As  to  form,  guides  may  be  divided  into  two  classes  :  angular 
guides  and  flat  guides.  Fig.  60  shows  an  angular  guide,  the 
pressure  being  applied  as  shown.  The  advantage  of  this  form 
is, 'that  as  the  rubbing  surfaces  wear,  the  slide  follows  down  and- 
takes  up  both  the  vertical  and  lateral  wear.  The  objection  to 
this  form  is  that  the  pressure  is  not  applied  at  right  angles  to  the 
wearing  surfaces,  as  it  is  in  the  flat  guide  shown  in  61.  But  in 
Fig.  6 1  a  gib,  A,  must  be  provided  to  take  up  the  lateral  wear. 
The  gib  is  either  a  wedge  or  a  strip  with  parallel  sides  backed 
up  by  screws. 

Guides  of  the  angular  forms  are  used  for  planer  tables.  The 
weight  of  the  table  itself  holds  the  surfaces  in  contact,  and  if 
the  table  is  light  the  tendency  of  a  heavy  side  cut  would  be  to 
force  the  table  up  one  of  the  angular  surfaces  away  from  the 
other.  If  the  table  is  very  heavy,  however,  there  is  little  danger 
of  this,  and  hence  the  angular  guides  of  large  planers  are  much 
flatter  than  those  of  smaller  ones.  In  some  cases  one  of  the 
guides  of  a  planer  table  is  angular  and  the  other  flat.  The  side 
bearings  of  the  flat  guide  may  be  omitted,  as  the  lateral  wear  is 
taken  up  by  the  angular  guide.  This  arrangement  is  un- 
doubtedly good  if  both  guides  wear  down  equally  fast. 


-  142- 

Fig.  62  shows  three  forms  of  sliding  surfaces  such  as  are  used 
for  the  cross  slide  of  lathes,  vertical  side  of  shapers,  the 
table  slide  of  milling  machines,  etc.  A  is  a  taper  gib  that 
is  forced  in  by  a  screw  at  D  to  take  up  wear.  When  it  is  neces- 
sary to  take  up  wear  at  B,  the  screw  may  be  loosened  and  a  shim 
or  liner  may  be  removed  from  between  the  surface  at  a.  C  is  a 
thin  gib,  and  the  wear  is  take-n  up  by  means  of  several  screws 
like  the  one  shown.  This  form  is  not  so  satisfactory  as  the 
wedge  gib,  as  the  bearing  is  chiefly  under  the  points  of  the 
screws,  the  gib  being  thin  and  yielding,  whereas  in  the  wedge 
there  is  complete  contact  between  the  metallic  surfaces. 

The  sliding  surfaces  thus  far  considered  have  to  be  designed  so 
that  there  will  be  no  lost  motion  while  they  are  moving,  because 
they  are  required  to  move  while  the  machine  is  in  operation. 
The  gibs  have  to  be  carefully  designed  and  accurately  set  so  that 
the  moving  part  shall  be  just  "tight  and  loose,"  i.  e.,  so  that  it 
shall  be  free  to  move,  without  lost  motion  to  interfere  with  the 
accurate  action  of  the  machine.  There  is,  however,  another  class 
of  sliding  parts,  like  the  sliding  head  of  a  drill  press,  or  the  tail 
stock  of  a  lathe,  that  are  never  required  to  move  while  the 
machine  is  in  operation.  It  is  only  required  that  they  shall  be 
capable  of  being  fastened  accurately  in  a  required  position,  their 
movement  being  simply  to  readjust  them  to  other  conditions  of 
work,  while  the  machine  is  at  rest.  No  gib  is  necessary  and  no 
accuracy  of  motion  is  required.  It  is  simply  necessary  to  insure 
that  jtheir  position  is  accurate  when  they  are  clamped  for  the 
special  work  to  be  done." 


-143  — 

VIII. 
AXLES,  SHAFTING.  AND  COUPLINGS. 


83.  Definitions  and   General   Equations.     [Unwin,   §   133, 
pages  208,  209  ] 

84.  Axles  loaded  transversely.     [Unwin,  §§  134,  135,  136.] 

85.  Shafts  under  Torsion  only.     [Unwin,  §§  137,  138  ] 

86  Shafts  subjected  to  combined  Torsion  and  Bending. 
[Unwin,  §  139.]  A  convenient  diagram  is  shown  in  Fig.  63  for 
determining  the  diameter  of  a  shaft,  of  solid  circular  cross-section, 
subjected  to  any  moment,  and  with  any  intensity  of  fibre  stress 
from  zero  to  15,000  Ibs.  per  sq.  inch.  This  diagram  can  be  used 
for  either  simple  bending  or  twining  moments,  or  for  combined 
bending  and  twisting  actions.  Its  use  in  connection  with  prob- 
lems involving  simple  twisting  moments  will  be  discussed  first. 

If  T  is  the  twisting  moment,  d  the  diameter  of  the  solid  circular 
shaft,  and  f  the  intensity  of  stress  in  the  most  strained  fibres, 

T=   —fd^.       Therefore,   for   a   given   diameter  of  shaft,    T  is 
16 

directly  proportional  to  f.  Thus,  if  d=j\."t  df3  — 64,  and 
T=  .196  X  64/=  12.57/1  If  /be  taken  as  10,000,  T=  125,700 
inch  Ibs.  In  Fig.  63,  if  ordinates  represent  moments  (to  the  scale 
"  A,"  of  10,000  inch  Ibs.  to  the  i")  ;  and  if  abscissas  represent 
intensity  of  stress  (to  the  upper  scale,  "  B,"  of  4,000  Ibs.  per  sq. 
inch  to  the  inch),  the  point  a  corresponds  to  T=  125.700,  /= 
10,000,  d  =  4".  As  the  moment  varies  directly  as  the  inteiiHly 
of  stress,  for  any  given  diameter  of  shaft,  the  relations  between 
corresponding  values  of  T  and  j  (for  a  4"  shaft)  will  be  repre- 
sented by  the  straight  line  through  the  point  a,  and  the  oiigin  O. 
In  a  similar  manner  straight  lines  through  the  origin  are  diawn 
for  other  shaft  diameters. 


-144  — 

To  determine  the  diameter  of  shaft  for  a  moment  oJ  90,000 
inch  Ibs,  with  a  fibre  stress  of  12,000  Ibs.  per  sq.  inch,  pass  along 
the  horizontal  through  the  point  marked  "9"  (or  7^=90,000) 
on  scale  "A,"  to  the  vertical  line  through  the  point  marked  "12" 
(or/=  12,000)  on  scale  "  B."  The  intersection  of  this  horizontal 
and  vertical  (t>)  lies  a  little  below  the  diagonal  marked  3.4  at  its 
outer  end  ;  or  the  shaft  should  be  about  3  37"  or  3^"  diameter  to 
give  a  stress  of  12,000  per  sq.  inch. 

The  oblique  line  nearest  to  the  point  located  in  the  last  ex- 
ample bears  three  figures,  viz.:  ".738  —  1.58  —  34,"  and  the 
other  diagonals  each  bear  three  separate  figures.  The  significance 
of  these  designations  will  be  explained  by  further  illustrations. 

If    T=^i\i    of  90,000,    or   9,000   inch    Ibs.    and  f=  12,000, 


'    (9°^  -  3.37  H-  *To  =  1.56"  - 

\       IO 


irf  \I2,OOOrr 

since  d  varies  as  the  cube  root  of  T  and  when  T—  90,000,  d  = 

3-37"- 

In   a  similar  way,   if  T^goo,  or  y-g-g-th   of  90000,   ^=3.37 


To  use  the  diagram  when  7^^900,  and  f^  12,000,  consider 
scale  "A"  as  representing  the  moment  in  100  inch  Ibs.;  pass 
along  the  horizontal  through  9  of  this  scale  to  the  vertical 
through  12  of  scale  "  B,"  as  before,  to  the  point  "  £,  "  and  take 
the  first  figure  borne  by  the  nearest  diagonal  (.732)  as  the  approxi- 
mate diameter  of  the  shaft  ;  or,  by  interpolation,  find  the 
diameter  =  .724". 

If  7^=9,000,  /=  12,000  ;  consider  scale  "  A  "  as  representing 
the  moment  in  1,000  inch  Ibs.,  and  read  the  middle  figure  on  the 
nearest  diagonal  (1.58)  as  the  required  approximate  diameter  of 
the  shaft  ;  or,  by  interpolation,  the  diameter  is  found  to  be  1.56''. 

If  the  moment  is  greater  than  130,000  (and  less  than  1,300,000) 
the  diagram  is  quite  as  applicable  as  for  smaller  moments.  Thus 
if  7^=900,000  and/=  12,000,  consider  scale  "  A  "  as  represent- 
ing the  moment  in  100,000  inch  Ibs.  The  horizontal  through  9 
of  scale  "  A  "  and  the  vertical  through  12  of  scale  "  B"  intersect 
at  "  b  "  as  before.  The  required  diameter  is  about  7.  24"  ;  because 


-  145  - 

the  diameter  was  found  to  be  about  .724  for  a  moment  of  900,  and 
it  must  be  10  times  as  great  for  a  moment  of  io~3  X  900  =  900,000. 
For/  =  12,000  with  a  moment  of  9,000,000  inch  Ibs.  (—  io~3  X 
9,000),  the  diameter  is  10  X  1.57  =  15.7",  etc.  It  thus  appears 
that  the  diagram  covers  all  moments,  without  being  of  such  im- 
practicable size  as  it  would  be  if  it  were  not  for  the  peculiar  de- 
signation of  the  oblique  lines  and  the  method  of  using  scale  "A." 
The  diagram  can  also  be  used  for  simple  bending  moments.  The 
expression  for  the  bending  moment  in  an  axle  of  solid  circular 
section  is 


while  the  expression  for  a  twisting  moment  is,  as  given  above, 


Therefore,  with  a  given  diameter  and  numerically  equal  fibre 
stress,  T  is  numerically  equal  to  2  M.  To  determine  d  for  given 
values  of  f  and  M,  multiply  M  by  2  to  get  the  equivalent  T,  and 
with  this  value  of  T,  proceed  as  in  the  former  examples. 

For  finding  the  diameter  appropriate  to  a  combined  bending 
and  twisting  moment,  the  equivalent  twisting  moment, 


Te  =  M  +  v'  M2  +  T2, 

is  to  be  determined  ;  see  art.  16.  This  equivalent  twisting  mo- 
ment is  readily  determined  from  the  diagram  by  use  of  scale  4<C" 
at  the  bottom  of  Fig.  63  and  a  pair  of  dividers,  when  the  simple 
bending  moment  (M)  and  the  simple  twisting  moment  (7^)  are 
given.  Example:  Suppose  M  =  30,000,  7^=40,000,  and/"  = 
13,000.  Consider  scales  "A"  and  "C"  to  measure  moments  in 
10,000  inch  Ibs.  Take  J/at  3  on  scale  "A"  with  one  point  of  the 
dividers,  and  7'  at  4  on  scale  "  C"  with  the  other  point  of  the 
dividers  ;  then  the  distance  between  3  on  scale  "A"  and  4  on  scale 
"  C  "  represents  \/M*  +  T2.  Swing  the  dividers  about  the  point 
at  3  on  scale  "A"  as  a  centre  until  the  other  point  reaches  scale 
"A"  (at  point  8)  ;  then  o  .  .  8  on  scale  "A"  =  o  .  .  3  +  3  .  .  8  = 
'  J/2  +  T2  =  Te.  With  the  value  of  7"e)  found  in  this  way, 


—  146  — 

proceed  as  in  case  of  a  simple  twisting  moment.  The  intersec- 
tion of  the  horizontal  through  8  (  TJ  and  the  vertical  through  13 
(/")  is  at  point  "  c."  Since  the  moments  correspond  to  units  of 
10,000  inch  Ibs.  on  scale  "A,"  the  largest  figures  of  the  diagonals 
are  to  be  read  in  determining  the  diameter.  The  point  "  c " 
therefore  indicates  a  diameter  of  between  3  o"  and  3.2"  ;  by  inter- 
polation the  diameter  is  taken  as  3.15".  By  computation  the 
diameter  is  found  to  be  3.14".  A  shaft  3-^-''  diameter  would  be 
proper  for  this  case. 

The  diagram  of  Fig.  63  is  equally  convenient  for  rinding  the  in- 
tensity of  stress  in  a  given  shaft  under  a  known  moment ;  or  the 
moment  on  a  given  shaft  corresponding  to  any  intensity  of  stress. 
Thus,  if  a  yf"  shaft  is  subjected  to  moment  of  1,000,000  inch  Ibs., 
consider  the  moment  units  as  100,000  inch  Ibs.,  pass  horizontally 
from  10  on  scale  "A"  to  a  point  slightly  below  the  diagonal 
marked  .776  (7.76"  diameter),  and  then  vertically  upward  to  scale 
"B,"  where  the  stress  is  read  as  about  10,950  Ibs.  per  sq.  inch 

If  it  is  required  to  find  the  twisting  moment  corresponding  to  an 
intensity  of  stress  of  9,000  Ibs.  per  sq.  inch  on  a  shaft  i£"  diame- 
ter ;  pass  vertically  downward  from  "  9  "  on  scale  "  B  "  to  a  point 
somewhat  above  the  diagonal  marked  "  i  .49  "  ;  then  horizontally 
to  5.9  on  scale  "A."  As  1.49  is  the  middle  number  on  the  diago- 
nal, the  moment  units  are  1,000  inch  Ibs.  ;  therefore  7^=5.9  X 
^ooo  =  5,900  inch  Ibs. 

87.  Mill  Shafting.     [Unwin,  §  140.]     See,    also,   eq.   (2),   (3) 
and  (4)  of  article  77  (Notes). 

88.  Hollow   Shafts.     [Unwin,   §   141.]     The    use   of  hollow 
shafts  not  only  reduces  the  weight  for  a  given  strength,  but  the 
removal  of  the  metal  from  the  core  of  a  steel  shaft  (or  of  the  ingot 
from  which  it  is  made)  very  greatly  increases  its  reliability  under 
repeated  application  of  stress. 

Shortly  after  a  steel  ingot  is  cast,  the  exterior  solidifies 
and  becomes  comparatively  cool  while  the  internal  portion  is 
still  fluid.  The  subsequent  contraction,  during  complete  cooling, 
is  much  less  in  the  exterior  walls  than  it  is  in  the  hotter  interior 
mass.  Unless  the  interior  is  "fed"  during  this  period,  it  will 


-  H7- 

be  less  dense  than  the  outer  portions  and  shrinkage  cavities  are 
apt  to  be  present  in  the  ingot.  Numerous  expedients  have  been 
adopted  to  reduce  this  evil,  among  which  is  ''fluid  compression," 
or  subjecting  the  ingot  to  heavy  pressure  immediately  after  it  is 
poured.  The  difficulty  is  not  entirely  overcome  by  such  means, 
however,  as  the  walls  of  large  ingots  become  too  rigid  to  yield  to 
the  pressure  before  the  interior  is  entirely  solidified.  The  ex- 
ternal walls  "  freeze,"  after  which  the  internal  shrinkage  is  made 
up  by  metal  flowing  from  the  upper  portion  toward  the  bottom  as 
long  as  any  of  it  remains  fluid.  This  leaves  a  shrinkage  cavity 
at  the  upper  end  of  the  ingot.  Gas  liberated  during  cooling 
collects  in  this  cavity  also.  The  result  of  these  two  actions  is  to 
form  what  is  called  the  "pipe,"  which  frequently  extends  to  a 
considerable  depth.  The  top  end  of  the  ingot  is  cut  off  and  re- 
melted,  but  this  does  not  insure  removal  of  all  of  the  pipe,  and  it 
involves  much  expense.  If  the  portion  cut  off  is  not  sufficient  to 
remove  all  of  the  pipe,  a  piece  rolled  or  forged  from  the  ingot  con- 
tains a  flaw  near  the  centre  which  is  drawn  out  into  a  long  crack 
if  the  ingot  is  worked  into  a  long  piece.  The  rolling  or  forging- 
may  squeeze  the  sides  of  the  cavity  together  so  that  it  is  not 
easily  detected  at  any  section,  but  as  this  work  is  done  at  a  tem- 
perature much  below  that  corresponding  to  welding,  the  defect  is 
not  removed.  This  flaw  is  more  or  less  irregular  or  ragged, 
hence  its  form  is  favorable  to  starting  a  fracture,  under  variations 
of  stress,  which  may  finally  extend  far  enough  to  cause  rupture. 
See  the  discussion  of  "  micro -flaws  "  and  gradual  fractures  on 
page  12. 

If  the  ingot  is  bored  out,  the  pipe  is  effectually  removed,  and 
the  metal  remaining  is  superior  to  that  of  a  solid  shaft.  It  will  be 
evident  that  casting  a  hollow  ingot  is  not  the  equivalent  of  boring 
out  one  which  was  cast  solid  ;  for  if  the  ingot  is  cast  hollow  the 
outer  and  inner  walls  cool  before  the  intermediate  mass  does,  and 
the  shrinkage  effect  takes  place  in  the  latter.  In  fact,  a  shaft 
made  from  a  hollow  ingot  is  worse  than  the  solid  shaft,  in  the  re- 
spect that  the  former  has  the  defective  material  nearer  the  outer 
fibres  where  the  stress  is  greater. 


—  148  — 

8g.  Span,  or  Distance  between  Bearings,  in  Lines  of 
Shafting.  [Unwin,  §§  142,  145].  The  deflection  of  a  beam  is 
proportional  to  the  load  upon  it,  to  the  cube  of  the  span,  and  in- 
versely as  the  moment  of  inertia  of  the  section.  With  a  shaft  of 
solid  circular  section,  the  transverse  load  due  to  its  own  weight  is 
proportional  to  the  square  of  the  diameter,  and  the  moment  of  in- 
tertia  is  proportional  to  the  fourth -power  of  the  diameter.  Hence, 
the  deflection  is  proportional  to  d1  L3  -r-  d\  or  to  Z,3  -r-  d'1 ;  and, 
for  a  given  limit  of  deflection,  L  -----  y$/d'\  which  is  eq.  (41)  of 
§  142  (Unwin). 

Kent's  Mechanical  Engineers'  Pocket-Book  (page  868)  says  : 
"The  torsional  stress  is  inversely  proportional  to  the  velocity  of 
rotation,  while  the  bending  stress  will  not  be  reduced  in  the  same 
ratio.  It  is,  therefore,  impossible  to  write  a  formula  covering  the 
whole  problem  and  sufficiently  simple  for  practical  application, 
but  the  following  rules  are  correct  within  the  range  of-velocities 
usual  in  practice.  For  continuous  shafting  so  proportioned  as  to 
deflect  not  more  than  y^  of  an  inch  per  foot  of  length,  allowance 
being  made  for  the  weakening  effect  of  key-seats. 


=  «    Pw:     — ,    L  =  t    (720 d\  for  bare  shafts 
R 


d  = 


for  shafts  carrying  pulleys,  etc. 


d  =  diam.  in  inches,  L,  =  length  in  feet,  R—  revs,  per  min." 

If  the  length  of  span  is  expressed  in  inches,  as  in  eq.  (41),  Un- 
win, Kent's  constants  correspond  to  y  =  108  for  bare  shafts,  and 
62  for  shafts  with  pulleys,  etc. 

It  is  well  to  check  by  the  formula  of  §  145,  Unwin,  when  the 
speeds  are  high. 

90.  Cold  Rolled  Shafting.  [Unwin,  §  143].  Shafting  which 
is  finished  by  a  cold  rolling  process,  instead  of  by  turning  it,  is 
largely  used.  It  is  very  true  as  to  cross-sections.  The  cold  work- 
ing raises  the  elastic  strength  of  the  shaft  ;  this  effect  being  great- 
est near  the  outside,  which  is  the  portion  subjected  to  the  highest 
stress. 


Cold  roiled  shafting  is  peculiarly  liable  to  be  "sprung  "  in  cut- 
ting key-ways,  etc.,  as  this  operation  removes  part  of  the  most 
compressed  metal,  and  thus  disturbs  the  condition  of  internal 
stress  in  the  material. 

91.  Expansion  of  Shafts.     [Unwin,  §  144]. 

92.  Crank  Shafts.     [Unwin,   §§  146,    147,    148].     An  analysis 
of  the  centre  crank  type  of  shaft,  similar  to  that  given  by  Profes- 
sor Unwin  (§  148)  for  the  side  crank   form,    is  outlined  below. 
Fig.  64  shows  a  centre  crank  shaft  of  the  form  commonly  used 
with  high  speed  engines.     It  will  be  assumed  that  the  two  fly-wheel 
pulleys  are  of  equal  weight,  and  that  the  member  is  symmetrical 
about  the  centre  of  the  crank  pin  ;  i.  e.,  that  a  =  a',  b  =  b' .     The 
forces  on  the  shaft  are  :  the  load  (/>)  on  the  crank  pin  due  to  the 
steam  pressure  on  the  piston  and  the  inertia  effects  of  reciprocat- 
ing parts  ;  the  gravity  action  on  the  wheels,   Wl  and  W^  the  belt 
pulls,   T2  +    7J ;  the  reaction  at  the  main  bearings,  ./?,  and  R^. 

Graphical  analysis,  in  conjunction  with  computations,  is  here 
very  convenient.  There  are  two  cases  to  be  considered  :  in  the 
first  the  power  is  delivered  by  a  belt  on  one  fly-wheel  ;  in  the 
other  the  power  is  divided,  part  being  delivered  by  each  wheel. 
The  former  results  in  the  more  severe  straining  actions  on  the 
shaft,  and  it  will  therefore  be  taken  for  discussion,  There  is 
always  one  element  of  uncertainty  in  designing  such  a  shaft  for 
an  engine  not  built  for  some  particular  service,  viz.  :  the  direction 
in  which  the  belt  will  lead  ;  but  this  is  not  of  the  first  importance, 
and  the  condition  shown  in  Fig.  64  represents  about  the  maximum 
straining  actions  for  a  horizontal  engine. 

The  force  transmitted  to  the  crank  pin  by  the  connecting  rod, 
(P)  varies  in  direction  with  the  angularity  of  the  rod,  but  it  is 
sufficiently  exact  to  consider  this  force  as  acting  parallel  to  the 
centre  line  of  the  engine,  or  perpendicular  to  the  plane  of  the 
paper  in  Fig.  64.  The  reaction  at  each  main  bearing  due  to  Pis 
Y-Z  P.  For  a  horizontal  engine,  this  force  is  represented  by  ^  P 
in  Fig.  64  (a)  ;  the  reaction  at  each  bearing  due  to  the  fly  wheel 
weight  is  IV1  =  H^,  since  the  crank  and  wheels  are  assumed  to  be 
symmetrically  disposed  relatively  to  the  bearings. 

L,et  IV represent  the  weight  of  one  wheel  in  Fig.  643.     Assum- 


ing  all  of  the  power  to  be  given  off  at  one  wheel  (as  W^,  the  re- 
sultant pull  from  the  belt  on  this  wheel  is  T,  +  T^  when  T.,= 
total  pull  on  the  tight  side  and  Tj  —  total  pull  on  the  slack  side 
of  the  belt.  If  this  resultant  belt  pull,  be  represented  by  T.t  -f  7] 
in  Fig.  64  (a),  the  total  reaction  at  the  left  hand  bearing  is 
equal  and  opposite  to  the  resultant  of  the  system  offerees  :  ^>  P, 
W,  and  T.,  4-  Tv  With  the  values  of  these  forces  and  the  direc 
tion  of  T.2  -f  7J  assumed  in  Fig.  64  (a),  the  resultant  force  acting 
at  the  wheel  (due  to  gravity  and  belt  pull)  is  approximately  hori- 
zontal ;  hence  its  line  of  action,  and  that  of  the  resultant  R ',  near- 
ly coincides  with  that  of  the  load  on  the  crank  pin.  This  condition 
gives  the  maximum  straining  action  on  the  shaft,  and  is  therefore 
a  safe  assumption.  Assuming,  then,  that  ^  and  R\  are  in  the 
same  plane  ;  take  a  section,  x  x,  through  the  centre  of  the  crank 
pin  and  compute  the  bending  moments  of  all  external  forces  to 
the  left  of  this  section  with  reference  to  it.  If  mn  (Fig.  64!)) 
is  the  bending  moment  of  the  force  S  at  section  x  x,  mng  is  the 
moment  diagram  of  this  force.  The  moment  due  to  ,5"  at  any  sec- 
tion is  given  by  that  vertical  ordinate  of  the  diagram  which  is  at 
a  distance  from  g  equal  to  the  distance  of  the  section  from  this 
same  point.  In  a  similar  way,  if  nq  is  the  bending  moment  due 
to  R^  about  xx,  nq/tis  the  moment  diagram  for  JRr  But  the 
moments  due  to  ^  and  R^  are  of  opposite  signs,  hence  the  dia- 
gram of  unbalanced  moments  is  the  shaded  area,  mqhgm  The 
twisting  moment  on  the  shaft  between  the  centre  of  the  pin  and 
the  wheel  is  equal  to  Pr.  Draw  the  rectangle  m  ijg  with  a 
height  mi  representing  this  moment  to  the  same  scale  used  for 
the  bending  moments.  Combine  the  unbalanced  bending  mo- 
ments for  various  sections  with  the  twisting  moments  (by  the 
methods  used  in  §§  147,  148  of  Unwiu)  and  the  diagram  klst\§ 
the  diagram  of  the  equivalent  bending  moment  (Me)  on  the  left 
hand  half  of  the  shaft.  This  equivalent  moment  is  seen  to  be  a 
maximum  at  the  centre  of  the  main  bearing,  and. the  diameter  of 
shaft  should  be  computed  for  this  maximum  moment  by  the 
equation 


The  diameter  at  any  other  section  may  be  computed  by  equation 
(i),  using  the  value  of  Me  appropriate  for  that  section.  The 
shaft  is  made  of  the  same  diameter  throughout  its  length,  or  it  is 
reduced  somewhat  from  the  outer  limits  of  the  bearings  to  the 
ends.  As  previously  stated,  the  crank  pin  is  frequently  made 
with  a  diameter  equal  to  that  of  the  main  bearings. 

It  will  bt;  noticed  from  Fig.  64  b  that  the  bending  moment  is 
zero  at  v.  Inasmuch  as  fracture  is  particularly  liable  to  start  at 
the  junctions  of  the  pin  or  the  shaft  with  the  crank  arms,  it 
appears  desirable  to  have  this  section  of  zero  bending  moment 
about  mid-way  between  these  two  junctions,  or  at  about  the 
middle  of  the  crank  arm.  In  a  vertical  engine,  the  maximum 
straining  action  occurs  when  the  belt  pull  is  vertically  downward. 
While  this  is  not  the  most  common  condition,  it  is  safest  to 
assume  it  for  the  general  case. 

93.  Couplings.     [Unwin,  §§  149  to  155]. 

The  standard  coupling  is  the  flange  coupling  shown  by  Figs. 
149  and  150  (Unwin).  Compression  couplings  are  also  much 
used  in  "lines  of  shafting.  The  form  shown  in  Fig.  151  (Unwin) 
is  largely  used  in  this  country  ;  as  is  also  a  simple  clamp  coupling 
similar  to  that  of  Fig.  152  (Unwin),  excepting  that  the  two  halves 
are  more  often  held  together  by  bolts  passing  each  side  of  the 
shaft  instead  of  by  the  bands  at  the  ends. 

94.  Clutches.     [Unwin  §§  156,  157]. 

The  prong,  or  jaw,  clutch  is  often  used  for  connecting  sections  of 
shafts  which  do  not  have  to  be  frequently  engaged  or  disengaged  ; 
or  when  this  does  not  have  to  be  done  when  either  shaft  is  running. 

The  form  of  friction  clutch  shown  in  Fig.  155  (Unwin)  is  not 
uncommon,  especially  for  engaging  a  loose  pulley  with  a  shaft ; 
though  the  more  usual  form  for  friction  cut-off  couplings  and 
clutch  pulleys  is  one  in  which  wooden  faced  jaws,  actuated  by  a 
system  of  levers  and  toggles,  clamp  a  ring  or  plate  to  engage  the 
clutch. 

95.  Universal  Coupling.     [Unwin,  §  158]. 

The  Hooke's  joint  is  a  useful  device  in  a  limited  way.  The 
nature  of  the  motion  transmitted  by  this  mechanism  is  discussed 
in  Kinematics  of  Machinerv. 


-152  — 

IX. 
FRICTIONAL  AND  TOOTHED  GEARING. 


95.  Frictional  Gearing.  [Unwin,  §§  177,  178,  183].  See, 
also,  Kinematics  of  Machinery,  (Barr),  arts.  51  to  55. 

Professor  Goss,  of  Purdue  University,  in  a  paper  read  before 
the  A.  S-  M.  E.  (Trans.,  Vol.  XVIII,  p.  102),  reported  the  re- 
sults of  some  tests  of  friction  wheels  from  which  the  following  ab- 
stract is  derived  : 

These  experiments  were  made  with  driving  wheels  having  fric- 
tion surfaces  of  compressed  straw  board,  and  followers  having 
turned  iron  faces. 

This  combination  gives  greater  resistance  to  slipping  than  two 
metallic  wheels.  The  softer  material  should  always  be  used  for 
the  face  of  the  driving  wheel,  in  order  that  the  wear  resulting, 
should  the  follower  stop  under  load,  will  be  distributed  around 
the  circumference  instead  of  being  concentrated  at  one  spot.  The 
above  mentioned  experiments  indicate  that : 

Slippage  increases  gradually  with  the  load  up  to  3  per  cent., 
but  when  the  slippage  is  between  the  limits  of  3  and  6  per  cent,  it 
is  apt  to  suddenly  "  increase  to  100  per  cent.  ;  that  is,  the  driven 
wheel  is  likely  to  stop." 

The  Coefficient  of  Friction  is  most  affected  by  slippage.  "  Its 
value  increases  with  increase  of  slip  until  the  latter  becomes  about 
3  per  cent.,  after  which  the  action  of  the  gearing  becomes  uncer- 
tain. With  a  slippage  of  2  per  cent.,  the  maximum  value  of  the 
coefficient  rises  above  25  per  cent."  A  value  of  20  per  cent,  is 
easily  attainable  with  wheels  of  8  inches  diameter  and  upward. 
The  coefficient  is  apparently  constant  for  all  pressures  of  contact 
up  to  150  to  200  Ibs.  per  inch  of  width  of  face  ;  but  it  decreases 
with  higher  pressures.  "  Variations  in  peripheral  speed  between 
400  and  2,800  feet  per  minute  do  not  affect  the  coefficient  of  fric- 
tion." 


-  153- 

Pressure  of  Contact,  The  power  transmitted  varies  directly 
with  the  pressure  of  contact  ;  the  coefficient  of  friction  remaining 
constant.  In  the  limited  duration  of  the  experiments,  no  indica- 
tion of  deterioration  of  the  surfaces  were  noted  under  a  pressure 
of  400  Ibs.  per  inch  of  face  ;  but  the  most  efficient  pressure  is  about 
150  Ibs.  per  inch  of  face. 

"  Horse  Power.  By  making  d  the  diameter  of  the  friction 
wheel  in  inches,  w  the  width  of  face  also  in  inches,  and  N  the 
revolutions  per  minute,  and  by  accepting  0.2  as  a  safe  value  for 
the  coefficient  of  friction,  and  a  pressure  of  150  pounds  per  inch  of 
width  of  face  as  the  pressure  of  contact,  the  horse-power  may  be 
written  as  : 

HP  =  150X0.2  X_^*<tXwXW  = 


33,000 

This  formula  is  believed  to  be  safe  for  friction  wheels  which  are 
eight  inches  or  more  in  diameter,  and  under  conditions  which 
make  it  possible  for  them  to  be  kept  reasonably  clean." 

96.  Wedge  Gearing,  or  "  V  Frictions."  [Unwin,  §  184]. 
See,  also,  Kinematics  of  Machinery,  art.  55. 

If  the  total  normal  pressure  on  all  the  wedge  surfaces  be  N,  the 
force  pressing  the  wheels  together  be  P,  and  the  tangential  force 
transmitted  be  7", 

N=P-i-sina  (i) 

in  which  a  is  half  the  angle  between  the  adjacent  faces  of  each 
groove  (or  wedge).     The  angle  2  a  is  often  about  40°. 

T<nN,  </*/>-=-  sin  a  (2) 

when  ju.  is  the  coefficient  of  friction  between  the  contact  surfaces. 
If  the  pitch  diameter  of  the  wheel  (the  diameter  to  the  middle 
of  the  depths  of  the  grooves)  be  D  feet,  and  the  revolutions  per 
minute  be  n,  the  power  transmitted  is, 


< 


33,000        33,000  sn  a 


(   } 


//./>.        --  (4) 

""•  10,500  sin  a 


p  =  10,500  sin  a  H.  P. 


-  154  - 

Mr.  Kent  states,  on  page  906  of  the  "  Pocket  Book,"  that  : 
"  The  value  of  p.  for  metal  on  metal  may  be  taken  at  .15  to  .20  ; 
for  wood  on  metal,  .25  to  .30."  The  number  of  grooves,  fora 
given  angle  a,  does  not  effect  the  relation  between  P  and  T.  But, 
for  a  given  face  of  wheel,  the  depth  of  grooves  is  increased  as  the 
number  is  decreased,  and  the  grinding  action  between  adjacent 
surfaces  is  proportional  to  the  depth  of  the  contact  faces  of  the 
"  Vs."  See  Kinematics  of  Machinery,  page  106. 

97.  General  Features  of  Toothed  Gearing.     [Umvin,  §§  185 
to  190,  inclusive]. 

The  relations  given  by  Professor  Unvvin  in  §§  191  to  209,  inclu- 
sive, are  usually  treated  in  courses  on  Kinematics,  and  are  there- 
fore omitted  from  these  Notes. 

98.  Power  Transmitted  by  Toothed  Gearing.     The  power 
transmitted  and  the  resulting  straining  actions  on  gears  and  their 
shafts  will  now  be  briefly  treated.     The  following  notation  is  used 
throughout  this  article.     See  Fig.  65. 

/y=the  normal  pressure  on  a  tooth,  which  acts  along  the  line 
connecting  the  pitch  point  of  the  gear  (#)  with  the  contact 
point  (£-). 

.P— the  tangential  component  of  the  preceding,  or  the  pressure 
which  acts  tangentially  to  the  pitch  circle. 

R^=  the  radius  of  the  pitch  circle. 

JV=  the  revolutions  per  unit  of  time. 

T—  the  turning  moment  on  the  gear  and  shaft. 

M=  the  bending  moment  on  the  shaft. 

If  the  foot-inch-minute  system  of  units  be  taken,  the  turning 

moment  in  inch  pounds  is, 

T=PX,         ..  (0 

and  the  energy  transmitted  in  inch  pounds  per  minute  is, 

E=2TtN.  PR,  (2) 

«qual  to  the  angular  velocity  times  the  rotative  moment,  or  to  the 
tangential  pressure  times  the  linear  velocity. 

2irNPR=   12  X  33,000  H.P. 

^=63,020^  (3) 


-155  — 

The  normal  pressure,  P'  =  P  sec  6  ;  but  the  arm  of  this  force 
(/>')  about  the  axis  of  the  shaft,  is  R1  ,  while  the  arm  of  the  tan- 
gential force  (/»)  is  R  ;  Fig.  65.  Since  R  =  Rt  sec  0,  T=  P'  K= 
PR,  or  the  turning  moment  is  the  tangential  pressure  times  the 
pitch  radius,  whatever  the  obliquity  of  action  and  the  actual  mag- 
nitude of  the  normal  pressure.  This  applies  to  all  systems  of 
gearing.  Referring  to  Fig.  66,  it  will  appear  that  the  reactions 
at  the  bearings  of  the  shaft  (Qt,  £?2),  hence  the  load  tending  to 
bend  the  shaft,  are  dependent  on  the  magnitude  of  P'.  If  the 
distances  of  the  bearings  from  the  gear  are  b  and  c  (as  in  Fig.  66), 


(4) 


The  bending  moment  equivalent  to  the  combined  bending  and 
twisting  action  is  Me  =  |  M  -\-  \  \S~M"  +  T*. 

The  obliquity  of  the  normal  pressure  at  the  teeth  is  thus  seen 
to  affect  the  bending  moment  on  the  shaft  and  the  total  pressures 
on  the  bearings,  but  it  does  not  affect  the.  twisting  moment  on  the 
shaft.  In  cycloidal  gearing,  the  obliquity  varies  from  a  maximum 
at  the  beginning  of  the  contact,  to  zero  when  the  contact  point 
lies  in  the  line  of  centres  ;  and,  during  the  arc  of  recess,  it  in- 
creases to  a  maximum  at  the  end  of  contact.  The  maximum 
value  of  the  angle  0  (Fig.  65)  is  about  22°  with  usual  forms  of 
cycloidal  gears.  When  0=  22°,  sec  0=  i.  08,  or  the  maximum 
normal  pressure  is  about  8  per  cent,  greater  than  the  tangential, 
rotative,  force.  The  obliquity  is  constant  throughout  the  arc  of 
action  in  involute  gears,  and  the  angle  0  is  usually  14^°  or  15°. 
If  0  =  15°,  sec  0=  1.035,  or  tne  normal  pressure  is  3^-2  per  cent. 
greater  than  the  tangential  force 

Mr.  Wilfred  Lewis,  in  the  American  Machinist  for  February 
28th,  1901,  advocates  increasing  the  obliquity  of  involute  gears  to 
22^°,  to  avoid  "  interference  "  of  the  teeth.  The  secant  of  this 
angle  is  1.082  ;  and  if  this  angle  were  adopted,  the  constant 
normal  pressure  would  be  about  equal  to  the  maximum  normal 


-  I56- 

pressure  with  the  cycloidal  gears'  of  usual  proportions.  This 
would  result  in  somewhat  greater  journal  friction,  but  the  com- 
pensating advantage  of  avoiding  interference  recommends  it  for 
pinions  of  few  teeth  and  with  moderate  loads.  This  greater 
obliquity  would  tend  to  increase  the  wear  on  the  teeth. 

98  Strength  of  Gear  Teeth.  [Unwin,  §§210,  211,  218.]  The 
assumptions  that  the  teeth  of  spur  gears  can  be  considered  as 
rectangular  prisms  in  determining  their  strength  is  not  satis- 
factory, especially  in  treating  of  pinions  with  a  low  number  of 
teeth.  Fig.  69  shows  four  gear  teeth  which  have  the  same  thick- 
ness at  the  pitch  line,  and  the  same  height.  The  tooth 
marked  (a)  is  one  of  an  involute  rack  ;  (b}  is  one  of  an  involute 
pinion  having  12  teeth  ;  *  (c)  is  one  of  an  epicycloidal  gear  having 
30  teeth  ;  (of)  is  one  of  an  epicycloidal  pinion  of  12  teeth. 

Mr.  Wilfred  Lewis,  of  Win.  Sellers  &  Co.,  seems  to  have  been 
the  first  to  investigate  the  strength  of  gear  teeth  with  due  regard 
to  the  actual  forms  used  in  the  modern  systems  of  gearing.  His 
work  was  originally  published  in  the  proceedings  of  the  Engi- 
neer's Club  of  Philadelphia.  January,  1893.  Numerous  formulas 
and  diagrams  have  since  been  devised  for  solving  the  problems 
connected  with  the  strength  of  gear  teeth. 

It  is  usually  intended  that  more  than  one  pair  of  teeth  shall  be 
in  action  at  all  times,  but  owing  to  unavoidable  inaccuracies  of 
form  and  spacing;  it  is  not  safe  to  depend  upon  -a  distribution  of 
the  load  between  two  or  more  teeth  of  a  gear.  It  is  safest  to  pro- 
vide sufficient  strength  for  carrying  the  entire  load  on  a  single 
tooth.  In  the  rougher  classes  of  work,  this  load  may  be  concen- 
trated at  one  edge  of  the  tooth,  as  indicated  in  Fig.  67,  (see  Un- 
win, §  21 1).  With  well  supported  bearings  and  machine  moulded 
or  cut  gears,  it  is  not  unreasonable  to  consider  the  load  as  fairly 
well  distributed  across  the  face  of  the  gear,  if  the  face  does  not  ex- 
ceed about  three  times  the  pitch  ;  see  Fig.  69.  The  obliquity  of 
action  gives  rise  to  a  crushing  action  on  the  teeth  (due  to  the 
radial  component  of  the  normal  force),  in  addition  to  flexural 

*  The  12  tooth  involute  pinion  may  have  its  teeth  weakened  by  a  correction 
for  interference  ;  but  it  is  usually  better  to  correct  the  points  of  the  mating 
wheel. 


stress  which  results  from  the  tangential  force.  This  crushing 
component  does  not  exceed  about  10  per  cent,  of  the  normal  pres- 
sure. Its  effect  is  to  reduce  the  tensile  stress  due  to  flexure,  and 
increases  the  compressive  stress.  As  cast  iron,  which  is  the  most 
common  material  for  gears,  is  far  stronger  in  compression  than  in 
tension,  this  radial  action  may  usually  be  neglected. 

Referring  to  Fig.  70,  it  is  seen  that  the  normal  force  P' ,  when 
acting  on  the  extreme  point  of  the  tooth,  produces  a  bending  mo- 
ment on  the  cross-section  a  a!  equal  to  P'  x'  =  P x.  Let  a  para- 
bola be  drawn  with  its  vertex  at  m  and  tangent  to  the  tooth  out- 
line curves  at  a  and  a' '.  This  parabola  represents  a  cantilever 
equal  in  strength  to  the  given  gear  tooth. 

In  examining  a  given  form  of  gear  tooth,  it  is  not  necessary  to 
actually  construct  the  parabola  in  order  to  locate  the  weakest  sec- 
tion with  practical  accuracy. 

The  strength  of  the  tooth  is  given  by  the  following  formula  : 


in  which  b  is  the  face  of  gear  teeth,  p  is  the  circular  pitch,  and  f 
is  the  intensity  of  stress. 

For  epicycloidal  gears  with  a  diameter  of  describing  circle  equal 
to  the  radius  of  a  12-tooth  pinion  of  the  same  pitch,  and  fillets 
equal  to  the  clearance  at  the  root  of  the  teeth,  Mr.  Lewis  gives,  as 
the  result  of  his  investigation,  the  formula, 

P=bpf(.\n  —  —  \  (i) 


in  which  P  is  the  load  per  tooth  in  pounds,  p  is  the  circular  pitch 
in  inches, /"is  the  working  stress  in  pounds  per  square  inch,  b  is 
the  face  of  the  gear  in  inches,  and  n  is  the  number  of  teeth.  Mr. 
L,ewis'  formula  is  convenient  for  determining  Pt  b,  p,  orf,  when 
the  number  of  teeth  (n)  is  known  ;  but  a  common  problem  in  de- 
sign is  to  determine  the  pitch  when  the  pitch  diameter  of  the  gear 
is  given,  and  the  number  of  teeth  is  unknown.  To  adapt  Mr. 
Lewis'  investigation  to  this  last  stated  problem,  the  following  is 
presented,  together  with  a  diagram  which  may  be  used  instead 


-158- 

of  numerical  computations  in  solving  specific  problems.  This 
diagram  was  published  in  the  Sibley  Journal  of  Engineering  for 
June,  1897,  and  also  as  a  discussion  of  a  paper  by  Professor  F.  R. 
Jones,  presented  before  the  A.  S.  M.  E.  (Vol.  XVIII,  page  766). 
The  first  step  in  the  derivation  of  the  new  formula  is  to  elimi- 
nate the  number  of  teeth  (n~)  and  to  introduce  the  pitch  diameter 
(D)  in  the  Lewis  expression. 

pn  =  T?D     .',     n  =  7rD 

.-.        P=bpf(.^--^)  =  bf(.^p-^/}       (a) 
If  the  load  per  inch  of  gear  face  is  />,  =  />  -r-  b, 

°  (3) 


(4) 

The  pitch  can  be  found  from  eq.  (4)  for  any  values  of  P},  D, 
and/",  when  the  face  of  gear  is  known  or  assumed.  A  common 
problem  is  as  follows  :  The  distance  between  two  shafts  and  their 
velocity  ratio  is  known,  required  the  pitch  of  spur  gears  connect- 
ing these  shafts  for  a  given  load  and  working  stress  on  the  teeth. 
The  centre  distance  of  the  shafts  and  the  velocity  ratio  fix  the 
diameters  of  the  gears.  The  face  of  the  gears  may  be  governed, 
approximately,  by  the  space  available,  or  it  may  be  assumed  by 
the  designer  upon  other  considerations.  To  illustrate  ;  suppose 
P=  15,000  Ibs. ,/^=  8,000  Ibs.  per  sq.  inch,  and  that  the  smaller 
gear  is  to  be  40"  diameter.  Also,  that  the  face  of  the  gear  may 
be  taken  as  6".  The  load  per  inch  of  face  is  />,  =  15,000  -f-  6  = 
2, 500  Ibs.  Hence, 


[.049-3-57X2,500^ 

\  8,000  X  40  ) 


/>  =  40.22-         .049 

\ 

The  diagram  (Fig.  71)  consists  of  a  series  of  curves  (one  for  each 
separate  pitch),   the  abscissas  of  which  (scale   "A")  represent 


-  i59  - 

diameters  of  gears  and  the  ordinates  (scale  "B"),  the  load  per 
inch  of  face,  for  a  stress  of  6,000  Ibs.  per  sq.  inch.  Any  other 
stress  coukl  have  been  taken  for  plotting  the  diagram,  and  any 
other  stress  may  be  used  in  solving  problems  by  it. 

To  illustrate  the  construction  of  the  diagram  for  one  curve,  take 
that  one  corresponding  to  2"  pitch,  and  let/=  6,000.  Substitu- 
ting 2  for/,  and  6,000  for  fin  eq.  (3), 

Pl  =  1,488  —  6-^  ;  hence,  when 

Z>  =  4.5,  Pl  =  o;  D=  10,  /»  =816;  D=2o,  Pl=  1,152;  etc. 

Plotting  the  corresponding  values  of  D  and  Pl  as  abscissas  and 
ordinates,  respectively,  the  curve  for  p  =  2"  is  drawn  through 
these  points.  The  other  curves  are  constructed  in  a  similar  way. 

If  any  one  stress,  as  6,000,  were  proper  under  all  circumstances, 
a  diagram  constructed  as  just  explained  would  be  sufficient ;  but 
different  materials  have  different  safe  working  stresses,  and  the 
stress  for  any  given  material  should  be  reduced  as  the  speed  or 
liability  of  shock  increases.  The  diagonal  stress  lines,  radiating 
from  the  lower  right  hand  corner,  are  provided  for  use  with  other 
stresses. 

The  upper  horizontal  scale  ("  C  ")  reads  from  right  to  left,  and 
its  divisions  correspond  to  those  of  scale  "  B." 

It  appears  from  eq.  (2)  that  the  load  on  the  tooth,  for  any  given 
pitch,  varies  directly  as  the  stress.  Or,  from  eq.  (3),  the  load  per 
inch  of  face  varies  directly  as  the  stress,  for  any  pitch.  Hence, 
for  any  pitch,  when  the  load  per  inch  of  face  and  the  stress  are 
fixed,  the  given  load  multiplied  by  6,000  and  divided  by  the  as- 
signed stress  is  the  equivalent  load  (for  this  pitch)  with  a  stress  of 
6,000  Ibs.  per  sq.  inch.  Thus,  a  load  of  2,000  Ibs.  per  inch  of 
face  and  a  stress  of  3,000  Ibs.  per  sq.  inch  requires  the  same  pitch 
as  a  load  of  2,000  X  6,000-^3,000  =  4,000  Ibs.  at  a  stress  of 
6,000.  The  pitch  for  a  gear  of,  say,  60"  diameter,  load  4,000  Ibs. 
.  per  inch  of  face,  stress  equal  6,000,  is  found,  from  the  diagram,  to 
be  about  7^"  ;  which  would  be  the  pitch  for  a  unit  load  of  2,000 
Ibs.  and  stress  =  3,000.  The  diagonal  lines  are  so  drawn  that  by 


—  160  — 

passing  vertically  downward  from  any  reading  (unit  load)  on  scale 
"  C"  to  one  of  these  diagonals  ;  thence  horizontal  to  scale  "  B," 
the  load  indicated  by  the  reading  on  "  B  "  will  be  the  equivalent 
load  for  a  stress  of  6,000  Ibs.  per  sq.  inch. 

To  illustrate  the  use  of  the  diagram,  take  P}  =  2,5oo,/=  8,<>co, 
Z>  =  40.  From  2,500  on  scale  "  C,"  pass  vertically  downward  to 
the  diagonal  marked  "^=8,000";  then  horizontally  to  point 
"<z",  on  the  vertical  rising  from  D=^^o"  (scale  "A").  The 
nearest  pitch  curve  is  that  marked  3".  This  curve  passes  some- 
what above  the  point  a,  hence  the  required  pitch  is  somewhat  less 
than  3". 

If  it  is  required  to  find  the  load  per  inch  of  face  for  a  gear  of 
given  diameter  and  pitch,  with,  an  assigned  stress,  start  at  the 
point  on  scale  "A"  corresponding  to  the  diameter  ;  pass  upward  to 
the  given  pitch  curve  ;  thence  horizontally  (right  or  left,  as  the 
case  may  be)  to  the  appropriate  stress  diagonal  ;  thence  upward 
to  scale  "  C,"  where  the  unit  load  is  read  off. 

If  the  diameter,  pitch,  and  unit  load  are  the  known  quantities, 
pass  upward  from  the  diameter  reading  on  scale  "A"  to  the 
proper  pitch  curve  ;  thence  horizontally  to  a  point  under  the  unit 
load  reading  on  scale  "  C,"  when  the  stress  is  found  by  interpola- 
tion between  the  adjoining  diagonals. 

Professor  F.  R.  Jones  presented  a  set  of  diagrams  (in  a  paper 
before  the  A.  S.  M.  K.  previously  mentioned  in  this  article)  which 
are  applicable  for  similar  solutions  to  those  discussed  above.  He 
based  his  work  upon  a  system  of  gearing  in  which  the  diameter  of 
the  generating  circle  is  equal  to  the  radius  of  a  15-tooth  pinion, 
and  the  fillets  are  drawn  with  a  "radius  equal  to  one-sixth  the 
space  between  the  teeth  at  the  addendum  circle."  With  this  lat- 
ter system  the  load  per  tooth  is  given  by  the  equation  : 


o.  106 -?  (5) 


p=pbf( 

The  system  used  by  Mr.  Lewis  gives  teeth  that  are  stronger  by. 
from  6  per  cent,  in  a  i5-tooth  pinion  to  17  per  cent,  in  a  rack. 

Of  course,  such  a  diagram  as  that  of  Fig.  71  could  be  drawn  for 
any  system  of  teeth  ;  but  the  particular  system  adopted  will  be 


—  161  - 

fairly  satisfactory  for  most  ordinary  cases,  witli  an  interchange- 
able system  of  epicycloidal  gearing. 

For  use  witli  a  limited  class  of  gears,  a  diagram  drawn  to  a 
larger  scale  and  covering  a  smaller  range  would  be  better  than 
the  one  here  shown  for  illustration. 

Involute  teeth  are  generally  of  considerably  stronger  form  than 
corresponding  cycloidal  teeth. 

It  may  be  noticed  that  the  different  pitch  curves  of  Fig.  71  all 
have  a  common  tangent  through  the  origin  o.  The  points  of  tan-, 
gency  correspond  to  the  diameters  of  gear  at  which  the  teeth  have 
radial  flanks  for  the  respective  pitches.  It  is  also  seen  that  the 
various  pitch  curves  intersect.  The  intersection  of  the  10"  and  8" 
curves,  for  example,  corresponds  to  a  diameter  of  about  41".  The 
interpretation  is  that  a  tooth  of  8"  pitch  (in  this  system)  is  as 
strong  as  one  of  10"  pitch,  when  the  diameter  is  41  inches.  The 
reasons  for  this  are  •  that  the  teeth  of  10"  pitch  are  longer,  and  a 
41"  gear  of  10"  pitch  has  a  number  of  teeth  —  13  — ,  while  an  8" 
pitch,  with  the  same  diameter,  gives  a  number  of  teeth  =  16  +  . 
For  diameters  less  than  41",  the  8"  pitch  is  the  stronger.  See  Fig. 
72.  With  a  diameter  of  22"  +  ,  the  10"  pitch  teeth  would  be  so 
"  under-cut  "  at  the  flanks  that  their  thickness  would  be  reduced 
to  zero,  except  for  the  fillets  at  the  bottom  (see  scale  "A,"  Fig. 
71)  ;  while  the  same  condition  is  not  reached  with  an  8"  pitch  un- 
til the  diameter  is  reduced  to  18". 

An  easily  remembered  relation  is  :  The  load  per  tooth  on  a  36- 
tooth  gear  equals  one-tenth  the  face  of  gear  in  inches  multiplied  by 
the  circular  pitch  in  inches  and  stress  in  pounds  per  square  inch.  A 
i2-tooth  pinion  will  carry  one-half  (£),  and  a  rack  one  and  one- 
fourth  (i£)  the  load  of  a  36-tooth  gear  with  a  given  stress. 

The  data  may  be  such  that  a  point  (corresponding  to  "  a  ")  will 
lie  to  the  left  and  above  all  of  the  pitch  curves  of  Fig.  71  ;  i.  <?., 
above  the  common  tangent  through  o.  If  the  same  data  were 
used  in  eq.  (4),  page  158,  an  imaginary  quantity  would  result. 
This  means  that  the  unit  load  taken  cannot  be  carried  with  the 
stress  and  diameter  assigned,  by  any  possible  pitch.  The  only 
recourse,  for  a  gear  of  the  given  diameter  and  total  load  (/>),  is  to 


—  162  — 

increase  the  face  and  thus  reduce  the  unit  load  (/*,),  or  to  use  a 
material  which  permits  a  higher  intensity  of  stress. 

It  is  often  advantageous,  with  gears  having  cast  teeth,  to 
"  shroud  "  the  smaller  gear  when  the  difference  in  the  diameters 
of  a  pair  of  gears  is  great ;  see  §  218  (Unwin).  When  the  pinion 
is  thus  shrouded,  its  strength  may  be  considered  as  in  excess  of 
that  of  its  unshrouded  mate,  and  the  computations  (or  strength 
can  then  be  applied  to  the  larger  gear.  Bvidently  both  gears  can 
not  be  shrouded  to  the  full  height  of  the  the  teeth  ;  but  both  may 
be  "  half-shrouded,"  i.  e.,  shrouded  to  the  pitch  circle.  This  lat- 
ter expedient  may  be  of  advantage  when  the  conditions  are  severe 
and  the  gears  are  of  nearly  equal  diameters. 

It  is  possible  to  make  shrouded  "  cut  "  gears  by  using  an  ''end- 
mill  "  for  the  cutter,  as  indicated  in  Fig.  73. 

It  should  be  remarked  that  the  teeth  of  the  smaller  gear  of  a 
pair  is  subject  to  the  greater  wear,  as  its  teeth  come  into  action 
the  more  frequently.  Hence  the  teeth  of  the  smaller  gear,  which 
are,  from  their  form,  initially  the  weaker,  have  their  strength 
reduced  more  rapidly  by  wear  than  those  of  the  mating  gear. 

99.  Limiting    Velocity    of    Toothed    Wheels.      [Unwin, 
§§  215,  216].     Small  toothed  gears  are  seldom  run  at  such  speeds 
that  they  are  in  danger  of  bursting  under  centrifugal  action  ;  but 
large  fly-wheel   gears  may  approach  a  dangerous  rim  velocity. 
The  safe  limit  is  discussed  by  Professor  Unwin  in  §  215. 

100.  Strength  of  Bevel  Gear  Teeth.     [Unwin,  §217].     Ac- 
cording to  Mr.  Lewis,  a  spur  gear  of  pitch  and  diameter  equal  to 
the   pitch   and  diameter  of  the  bevel  gear  at  the  large  ends  of  the 
teeth,  is  stronger  than  the  bevel  gear,  in  the  ratio  of  D  to  d  ;  when 
D  is  the  pitch  diameter  at  the  large  end,  and  d  the  pitch  diameter 
at  the  small  end.     This  assumption  may  be  made  except  when 
the  face  of  the  bevel  gear  teeth  is  excessively  long.     Using  the 
notation  above,  and  other  notation  as  in  art.  98, 

(0 

(2) 


-^.048—3- 


-  i63- 

Conipare  eqs.  (3)  and  (4)  of  art.  98. 

It  is  more  difficult  to  insure  the  uniform  distribution  of  load 
along  the  elements  of  bevel  gears  than  in  spur  gears.  For  this 
reason  the  length  of  face  should  not  be  unnecessarily  long  in 
bevel  gear  teeth. 

101.  Width  of  Face  of  Gears.    [Unwin,  §  219].    The  strength 
and  the  durability  of  gear  teeth  increase  with  increase  efface,  if 
the  shafts  are  in  perfect  alignment.     The  difficulty  of  securing 
Uniform   distribution   of  the   load   along  the  contact  element  in- 
creases as  the   face  becomes  greater.     This  imposes  a  practical 
limit  to  increase  of  tooth  face.     The  space  available  for  the  gears 
and  the  "over-hang"  (if  the  gears  are  on  the  projecting  ends  of 
shafts)  sometimes  fix  the  limit  of  face. 

The  space  available  does  not  usually  prevent  extending  the  face 
of  bevel  gear  teeth  in  the  direction  toward  the  apex  of  the  cone, 
the  large  diameter  of  the  gear  being  determined.  However,  little 
is  gained,  and  serious  difficulty  is  encountered  by  going  to  an  ex- 
treme in  this  respect.  The  portion  of  the  teeth  added  by  exten- 
sion toward  the  intersection  of  the  shafts  is  of  small  strength,  rel- 
atively to  a  similar  length  efface  near  the  large  ends  of  the  teeth. 
And  the  difficulty  of  securing  uniform  distribution  of  load  along 
the  teeth  elements  (mentioned  in  the  preceding  article)  is  increased 
by  such  extension.  If  the  bevel  gear  is  cut  by  the  ordinary  mill- 
ing cutter  process,  the  tooth  elements  do  not  converge  accurately 
toward  the  apex  of  the  pitch  cone,  and  this  error  increases  with 
the  length  of  face.  With  cast  gears,  or  gears  with  planed  or 
"  moulded  "  teeth,  this  last  objection  does  not  hold. 

102.  Rims  of  Gears.     [Unwin,  §  220].     The  thickness  of  the 
rim  of  a  gear  is  commonly   about  equal   to  the  thickness  of  the 
teeth  at  the  roots.     This  proportion   is  usually  desirable  on  the 
score  of  securing  good  castings,  though  it  may  be  departed  from. 
If  the  distance  between  arms  is  so  great  that  more  rigidity  of  rim 
is  desirable  (which  is  generally  the  case)  the  rim  is  ribbed,  as  in- 
dicated in  Fig.  236  (Unwin).     Small  gears  are  often  made  solid, 
that  is,  of  the  same  thickness  from  rim  to  hub.     A  plate  gear  is 
used  if  the  diameter  is  rather  too  great  for  a  solid  gear,  but  not 


—  164  — 

great  enough  to  make  the  use  of  arms  desirable  ;  that  is,  a  web, 
or  flat  plate,  extends  along  the  mid  plane  of  the  gear  from  lim  to 
hub. 

103.  Arms  of  Gears.  [Unwin,  §  221  ]  In  small  gears  the 
arms  are  usually  proportioned  largely  by  the  judgment  of  the 
designer.  The  thickness  of  metal  in  the  arms  should  not  differ 
greatly  from  the  adjacent  thickness  at  the  rim  and  at  the  hub,  for 
if  the  casting  does  not  cool  quite  uniformly  the  shrinkage 
stresses  often  exceed  those  due  to  the  load  transmitted.  Professor 
Unvvin  gives  three  methods  of  computing  the  arms.  The  first, 
in  which  the  ratio  h  -i-u  is  assumed,  may  be  used,  but  it  is  per- 
haps better  in  the  usual  case  to  make  the  thickness  a  and  /8 
(Fig.  241,  Unwin)  about  equal  to  the  rim  thickness,  or  about 
equal  to  ^2  the  pitch.  If  a  be  taken  as  .5 p  in  eq.  (14), 

«=I2^ 

vfP 


=  I12?!? 

\    vfp  ' 


The  second  method  given  in  Unwin  is  based  upon  the  assump- 
tion that  the  gear  tooth  can  be  treated  as  a  rectangular  prism  in 
considering  its  strength,  which  is  not  in  accordance  with  the  pre- 
ceding work  on  strength  of  gear  teeth.  • 

The  third  method  given  by  Unwin  seems  best  for  the 
usual  case,  but  eq.  (i),  above,  may  be  used  instead  of  eq.  (18)  of 
Unwin.  The  text  ot  §221  should  be  read;  but  eqs.  (16),  (17) 
and  (18)  may  be  omitted. 

104.  Hubs  of  Gears.  [Unwin,  §  222].  It  is  quite  common 
to  make  the  diameter  of  the  hub  twice  the  diameter  of  the  shaft  ; 
that  is,  the  thickness  of  metal  in  the  hub  is  equal  to  the  radius  of 
the  shaft.  In  case  of  a  light  gear  on  a  large  shaft,  this  rule 
would  give  excessive  thickness  of  metal  ;  that  is,  more  than 
strength  demands,  and  also  a  difference  betweeii  hub  and  rim  thick- 
nesses which  would  tend  to  unnecessarily  increase  the  shrinkage 
stresses.  For  such  conditions  as  these,  the  hub  should  be  lighter 
than  the  common  rule  indicates. 

It  is  not  usual,  in  this  country,  to  enlarge  the  shaft  where  it 
passes  through  the  gear  ;  the  preference  being  for  a  straight  shaft 


-i65- 

^ 

large  enough  to  permit  keyseating  without  undue  weakening  of 
the  shaft.  This  practice  usually  saves  more  in  shop  work  than 
the  opposite  course  would  save  in  material  ;  though  there  are  ex- 
ceptions to  this  rule. 

The  term   "  nave"   is    the    British  name  for  what  is  usually 
called  the  hub  in  this  country. 

105.  Weight  of  Toothed  Gearing.     [Unwin,  §  223]. 

106.  Effiicency    of    Spur  Gearing.     The  experimental  data 
on    the    efficiency    of    spur  gears    is   apparently     very    meagre. 
Probably  the  best  available  data  are  those  obtained  by   Mr.   Wil- 
fred Lewis,  for  details  of  which  see  Trans.  A.  S.  M.  E.  Vol.  VII, 
page  273.     His  investigation  was  made  with  a  cut  spur  pinion  of 
12  teeth  meshing  with  a  gear  of  39  teeth.     The  pitch   was  1%" 
and  the  face  was  3^".     The  load  was  varied  from  430  Ibs.   to 
2,500  Ibs.  per  tooth,  and  the  peripheral  speed  ranged  from  3  feet  to 
200  feet  per  minute.     The  measurements  included  the  friction  at 
the  teeth  and  the  friction  of  the  two  shafts.     The  efficiency,    as 
observed,   varied   from  90  per   cent,  at  a  velocity   of  3   feet  per 
minute  to  over  98  per  cent,   at  300  feet  per  minute.     It  appears 
that  the  friction  at  the  teeth  is  a  small  part  of  the  loss,  with  good 
cut  gears  ;  the  greater  portion  of  the  loss  being  at  the  journals. 
This  latter  is,  however,   a   necessary  loss  incident   to  the  use  of 
gearing. 

107.  Helical  or  Twisted  Gearing.     [Unwin,  §§  224  to  226]. 

108.  Screw  Gearing.     [Unwin,  §§  227-228].     The  expression 
for  the  efficiency  (77)  of  screw  gears  [eq.   (26),   Unwin]  can    be 
reduced  to  the  following  : 

_  tanfl  (i  —  /utan  6) 

»= 


in  which  B  is  the  inclination  of  the  pitch  line  helix  to  a  plane 
perpendicular  to  the  axis  ;  and  p  is  the  coefficient  of  friction. 
This  does  not  include  the  friction  at  the  thrust  bearing,  which  is 
often  an  important  item  in  a  worm  and  wheel  mechanism.  The 


—  166  — 

following  is   an   approximate  expression    for  the  efficiency, 
cluding  the  thrust  bearing.* 


tan  6(\  —  a  tan  6) 

11  =  —- 


The  above  formulas  are  based  upon  the  assumption  that  the 
worm  teeth  correspond  to  the  threads  of  a  square  threaded  screw. 
As  this  is  not  the  case,  it  would  be  natural  to  expect  the 
real  efficiency  to  be  somewhat  lower  than  these  expressions 
give.  The  experiments  of  Mr.  Lewis  show  a  very  satisfactory 
agreement  with  the  latter  formula.  See  Trans.  A.  S.  M.  E.  Vol. 
VII,  p.  273  ;  also  article  by  Mr.  F.  A.  Halsey,  American  Ma- 
chinist for  Jan.  131!)  and  2oth,  1898.  An  abstract  of  these  last 
named  articles  will  be  found  in  Ketit's  Mechanical  Engineer's 
Pocket-Book,  page  1078. 

The  frictional  work  at  the  teeth  of  a  spiral  gear  is  proportional 
to  the  velocity  of  rubbing  ;  hence  the  efficiency  increases  as  the 
diameter  decreases,  for  a  given  rotative  speed.  An  examination 
of  eqs.  (2)  or  (3)  shows  that  the  efficiency  increases  very  rapidly 
with  increase  of  the  inclination  (0)  at  low  angles.  The  maxi- 
mum efficiency  if  n=  .05  is  at  an  angle  of  about  43^°  (neglect- 
ing the  thrust  bearing)  or  at  about  53°  (including  the  thrust 
bearing).  With  an  increase  of  the  angle  beyond  these  limits,  the 
efficiency  falls  off.  The  smaller  inclinations  correspond  to  single 
threaded  worms,  or  at  least  to  worms  of  only  a  few  threads. 
The  higher  angles  are  obtained  with  spiral  gears  of  several 
threads.  With  a  value  of  0  much  greater  than  45°,  the  other 
gear  of  the  pair  approaches  more  nearly  to  the  special  form  of 
spiral  gear  commonly  called  a  worm,  because  the  inclinations  of 
the  helices  of  the  pair  are  complimentary  when,  as  is  most  com- 
mon, the  shafts  are  at  right  angles. 

If  0  =  60°  for  one  of  the  gears,  the  inclination  of  the  helix  of 
the  mating  gear  would  be  30°,  if  the  axes  are  at  right  angles.  It 


*  Equation  (2)  is  based  on  these  assumptions  :  that  the  mean  diameter  of 
the  thrust  collar  is  equal  to  the  pitch  diameter  of  the  worm  ;  and  that  the 
coefficient  of  friction  is  the  same  at  the  teeth  and  thrust  collar. 


—  i67  — 

would  therefore  be  reasonable  to  expect  about  the  same  efficiency 
at  0  =  30°  and  0  =60°.  Neglecting  the  friction  at  the  thrust 
bearing,  this  would  be  substantially  the  result. 

A  ball  bearing  step  has  been  used  to  good  purpose  in  reducing 
the  step  friction.  The  objection  to  this  device  is  the  danger 
of  cutting  the  ball  races  under  heavy  loads. 

109.  Construction  of  Screw  Gearing.  [Unwin.  §§  230  to 
236.]  The  usual  method  of  making  accurate  worm  wheels  is  to 
use  a  "  hob  ",  which  is  a  milling  cutter  of  similar  general  form  to 
the  worm  which  is  to  mesh  with  the  wheel.  See  Kinematics  of 
Machinery,  page  168.  For  this  reason  it  is  seldom  necessary  to 
make  an  accurate  drawing  of  a  worm.  However,  worms  with 
cast  threads  are  still  used  in  some  classes  of  heavy,  rough  work, 
and  the  method  of  laying  out  the  teeth  is  fully  treated  in  §§  233 
to  236  (Unwin). 

no    Strength  of  Worm  Wheels.     [Unwin,  §  236.] 


—  i68  — 

X. 

BELT    TRANSMISSION. 


in.   Materials  of  Belts.     [Unwin,  page  369.] 

112.  Velocity  Ratio  in  Belt  Transmission.    [Unwin,  §237.] 

113.  Resistance  to  Slipping  of  Belts.    [Unwin,  §§  241,  242.] 
The  equations  given  in   §  241   are  very  generally  used  ;  but  the 
assumptions  on  which  they  are  based  are  only  approximations  to 
the  conditions  of  operation.     This  theory  considers  slipping  as 
impending.     It   is  probable  that  slippage   occurs  whenever  the 
belt  transmits  power,  and  that  the  rate  of  slippage  gradually  in- 
creases with  the  effective,  or  unbalanced,  belt  pull,  to  a  certain 
point  at  which   the  belt'  either  runs  off  the  pulleys  or  slips  so 
much  that  it  fails  to  drive.     The  coefficient  of  friction  is  not  con- 
stant, but  it  increases  with  slippage. 

114.  Tensions  in  an  Endless  Belt.     [Unwin,  §§  243,  244.] 
The  assumption  that  the  sum  of  the  tensions  on  the  two  sides 

of  the  belt  remains  constantly  equal  to  the  sums  of  the  initial 
tensions  has  been  proved  to  be  in  error  by  the  experiments  of 
Messrs.  Lewis  and  Bancroft,  (Trans.  A.  S.  M.  E.,  Vol.  VII,  page 
549).  The  equations  given  by  Professor  Unwin  are,  therefore, 
not  exact  ;  though  they  are  convenient  for  many  computations, 
and  are  those  which  have  generally  been  accepted.  A  brief 
abstract  of  Mr.  Lewis's  paper  is  given  in  art.  122,  below. 

115.  Strength  of  Leather   Belting.     [Unwin,  §  245.]     The 
practice  of  Wm.  Sellers  &   Co.,   as  reported   by   Mr.    Lewi-   in 
the  transactions  of  the   A.  S.   M.  E.,  Vol.  XX,  page    152,  is  to 
take/=4oo8  for  cemented   belts,   (with   no  laced  joints);  and 

_/"==  275  8  for  laced  belts.  In  this  relation,  /is  the  tension  on  the 
tight  side  of  the  belt  in  Ibs.  per  inch  of  width,  and  8  is  the  thick- 
ness of  the  belt  in  inches  ;  as  in  §  245  (Unwin).  A  still  lower 
stress  increases  the  life  of  the  belt. 


-i69- 

1 16.  Width  of  Belt  for  a  given  Stress.     [Unwin,  §  246.] 

117.  Horse-power   per   inch   of  Belt   Width.      [Unwin,  § 

247-] 

118.  Rough  Calculations  of  Belt  Width.     [Unwin,  §  248.] 

1 19.  High  Speed  Belting.     [Unwin,  §  249.]     It  appears  that 
the  effect  of  a   high  speed  of  belt  frequently  tends  to  reduce  the 
adhesion,  rather  than  to  increase  it  by  formation  of  a  partial  vac- 
cuum.     There   are  two  causes   for   this   reduction   of  adhesion. 
One  of  these,  the  centrifugal  action  due  to  the  weight  of  the  belt, 
will  be  treated  later  ;  the  other  cause  is  the  adhesion  of  air  to  the 
belt.     This  latter  action  tends  to  carry  a  film  of  air  between  the 
belt  and  the  pulley,  as  the  oil  film   is  carried  into  the  space  be- 
tween  a  journal  and  its  bearing.     The  viscosity  of  oil  is  much 
greater  than  that  of  air  ;  but,  on  the  other  hand,   the  velocity  of 
the  belt  is  very  high,  compared  with  that  of  a  journal.     To  allow 
this   entrained    air   to   escape,    belts   are   sometimes    perforated, 
usually  with  oblong  holes  in  order  to  avoid  excessive  weakening 
of  the   belt.     This  practice  tends   to  increase  the  stretch  of  the 
belt.     Occasionally  the  pulley  rim  is  perforated   to  allow  the  es- 
cape of  the  air  without  reduction  of  the  cross-section  of  the  belt. 

120.  Influence  of  Elasticity  of  the   Belt.     [Unwin,   §  250.] 
The  slippage  of  belts,    as   measured  in   belt  tests,  is  made  up  of 
creeping  due  to  the  stretch  of  the  belt,   and  rt-al   .-lippage  of  the 
belt  on   the  pulleys  ;  both  of  which  occur  when  power  is  trans- 
mitted. 

121.  Effect  of  Centrifugal  Action.     [Unwin,  §  251.] 

122.  Recent    Investigations    of    Belt    Transmission.      A 
number   of  important   investigations  of  belt   transmission    have 
been  reported  to  the  American  Society  of  Mechanical   Engineers. 
See  the  following  papers  in  the  transactions  of  the  Society,    by  : 
Mr.  A.  F.  Nagle,  Vol.  II,    page  91.     Professor   G.    Lanza,   Vol. 
VII,  page  347.     Mr.  Wilfred   Lewis,   Vol.    VII,   page  549.     Mr. 
F.   M.   Taylor,    Vol.   XV,   page  204.     Professor  W.  S.  Aldrich, 
Vol.  XX,  page  136.     Abstracts  of  some  of  these  papers,   as  well 
as  other  valuable  data,  are  given  in  Kent's  Mechanical  Engineers' 
Pocket-Book,  pages  876  to  887. 


Mr.  Lewis'  paper  gives  the  results  of  and  conclusions  from 
the  very  careful  tests  conducted  by  himself  and  Mr.  Bancroft  for 
William  Sellers  &  Co.  The  apparatus  used  by  them  was  after- 
ward presented  to  Sibley  College,  and  is  now  used  by  the  De- 
partment of  Experimental  Engineering.  These  tests  indicate 
that  with  open  or  straight  belts,  the  journal  friction  is  the  prin- 
cipal resistance  at  moderate  speeds,  and  that  air  resistance  be- 
comes appreciable  at  high  speeds.  With  crossed  belts,  the  rubbing 
together  of  the  sides  of  the  belt  in  crossing,  and  the  resistance  at 
the  point  where  the  belt  leaves  the  pulley  are  often  sources  of 
considerable  loss.  The  bending  of  the  belt  around  the  pulleys 
did  not  result  in  an  appreciable  loss  of  energy  ;  narrow  thick  belts 
being  as  efficient  as  wide  thin  belts,  apparently.  The  rate  of 
the  strain  in  leather  decreases  with  the  stress,  instead  of  increas- 
ing as  in  the  case  of  ductile  metals.  This  property  is  similar  to 
that  possessed  by  soft  rubber,  and  it  becomes  very  apparent  in 
the  latter  material  when  a  common  rubber  band  is  stretched  out 
by  the  fingers.  As  a  result  of  this  property  of  leather,  the  sum 
of  the  tensions  on  the  two  sides  of  the  belt  is  not  constantly 
equal  to  the  sum  of  the  initial  tensions.  The  reason  for  this  is 
as  follows:  When  the  belt  transmits  power,  the  tension. is  in- 
creased on  the  driving  side  and  is  decreased  on  the  slack  side. 
A  given  reduction  of  tension  on  the  slack  side  tends  to  shorten 
that  side  of  the  belt  more  than  the  tight  side  is  increased  in  length 
by  the  same  increase  of  its  tension  ;  consequently  the  resultant 
effect  of  transmission  of  power  by  a  belt  tends  to  shorten  its 
length  as  a  whole  or  to  increase  the  sum  of  the  tensions. 
Suggestion  : — Place  a  rubber  band  over  the  fingers  of  the  two 
hands  and  stretch  it  moderately  ;  then  twist  one  of  the  hands  in 
either  direction  and  the  increase  of  force  tending  to  bring  the 
hands  together  will  be  apparent. 

In  case  of  a  long  horizontal  belt,  the  increase  in  the  sum  of  the 
tensions  is  still  further  augmented  in  driving,  because  the  tension 
on  the  slack  side  (with  a  proper  initial  tension  on  the  belt)  is 
largely  due  to  the  sag  of  the  belt  from  its  own  weight,  and  thus 
the  tension  on  the  slack  side  tends  to  remain  nearly  constant, 


while  the  tension  on  the  tight  side  increases  with  the  power 
transmitted,  at  a  given  belt  speed.  It  appears  that  the  sum  of  the 
tensions  on  the  two  sides  when  driving  may  exceed  the  sum  of 
the  initial  tensions  by  about  33  per  cent. in  vertical  belts,  and,  in 
horizontal  belts,  the  increase  may  be  limited  only  by  the  strength 
of  the  belt.  In  addition  to  the  causes  just  discussed,  the  tensions 
in  both  sides  of  the  belt  are  increased  by  the  centrifugal  action 
due  to  the  mass  of  the  belt.  This  latter  cause  increases  the 
stress  in  the  belt  and  decreases  adhesion  between  the  belt  and  the 
pulley,  but  it  does  not  increase  the  loads  on  the  shafts  which  pro- 
duce pressure  at  the  bearings  and  flexure  of  the  shafts. 

The  slippage  of  the  beft,  under  good  conditions,  may  be  as- 
sumed at  about  2  percent.;  about  i  percent,  being  "creeping" 
due  to  the  elasticity  of  the  material,  and  the  remainder  being 
true  slip,  or  sliding  of  the  belts  on  the  pulleys. 

The  coefficient  of  friction  increases  with  the  velocity  of  sliding, 
up  to  the  point  at  which  the  belt  tends  to  leave  the  pulley.  A 
given  actual  velocity  of  sliding  represents  a  smaller' percentage  of 
slip  as  the  belt  speed  increases  ;  hence  the  actual  velocity  of 
sliding  and  the  coefficient  of  friction  may  be  increased  at  higher 
speeds,  while  the  percentage  of  slippage  is  reduced.  The  slip- 
page may  become  20  per  cent,  or  more  before  the  belt  leaves  a 
crowned  pulley  ;  but  much  lower  slippage  is  desirable  on  the 
score  of  efficiency  and  durability  of  the  belt.  If  the  tension  on 
the  slack  side  is  too  low,  the  slippage  becomes  excessive  ;  on  the 
other  hand,  too  g.reat  tension  on  the  belt  results  in  unnecessary 
increase  of  the  journal  friction,  and  excessive  wear  of  the  belt. 

Either  of  these  extremes  reduces  the  efficiency  of  transmission. 
It  appears  that,  with  favorable  conditions,  the  efficiency  of  trans- 
mission by  an  open  belt  may  be  as  high  as  97  per  cent. 

The  coefficient  of  friction   of  the  belt  on  the  pulley  may  be 
taken  at  about  .40,  except  for  dry  belts  at  slow  speeds.     With  an 
arc  of  contact  of  180°,  the  coefficient  of  friction  was  found  to  be 
about : 
fi— .25  for    dry  oak  tanned  leather  at  a  speed  of  90  feet  per 

minute,  and 
p  =  1.38  for  a  very  flexible  rawhide  belt  at  800  feet  per  minute. 


—  172  — 

A  value  of  p.  =  i.oo  is  quite  possible,  though  not  to  be  depended 
upon  for  ordinary  working  conditions. 

Mr.  Nagle  gives  the  following  formula  for  the  power  trans- 
mitted, having  due  regard  to  the  effect  of  centrifugal  action  in 
increasing  the  tension  : 

/f.P.=  CVp*(J-.oi2  F2)-55o  (i) 

in  which  V—  the  velocity  of  the  belt  in  feet  per  minute,  ft  =  the 
width,  of  the  belt  in  inches,  8  =  the  thickness  of  the  belt  in  inches, 
/=  the  working  strength  of  leather  in  Ibs.  per  sq.  inch  cross- 
section,  and  C=a  constant  determined  by  the  formula, 


C=  j  —  jo"- 

when  //.=  the  coefficient  of  friction,  and  6       the  arc  of  contact  in 
degrees. 

Solving  eq.  (i)  for  the  width  of  belt, 


—  .oi2Fz) 

The  power  transmitted  by  a  belt  increases  directly  with  the 
belt  velocity,  except  for  the  effect  of  the  centrifugal  action.  The 
stress  due  to  centrifugal  action  increases  with  the  square  of  the 
velocity  ;  hence,  for  a  given  value  of  the  working  stress  (_/"), 
there  is  a  li  nitin^  velocity  at  which  the  greatest  power  is  trans- 
mitted. Differentiating  eq  (i)  with  reference  to  H.P.  and  Vy 
considering  the  other  quantities  as  constant,  placing  the  differen- 
tial coefficient  equal  to  zero,  and  solving  for  V,  it  will  be  found 
that 

V  =  */28/=  5.  29  v/7  (4) 

If/be  taken  at  400  Ibs.  per  sq.  inch  of  section  for  cemented 
belts  (see  art.  115),  V=  5.29  X  20  =  105.8  feet  per  second,  or 
6350  feet  per  minute.  If  f  is  taken  at  275  Ibs.  per  sq.  inch  for 
laced  belts,  V=.  5.29  X  16.  6=  87.8  feet  per  second,  or  5270  feet 
per  minute.  It  is  often  necessary  to  run  belts  at  much  lower 
speeds  than  these  ;  but  it  is  not  economical  to  exceed  these  limits. 
A  belt  speed  of  a  mile  a  minute  may  be  taken  as  about  the 


-  173- 

econoinical  limit  ;  and  it  so  happens  that  this  is  also  about  the 
limit  of  s.ifety  for  ordinary  cast  iron  pulley  rims.  See  §  259, 
(Unvvin).  For  durability  combined  with  efficiency,  a  speed  of 
3000  to  4000  feet  per  minute  may  be  taken  as  a  fair  belt  speed. 
It  was  stated  above,  that  the  coefficient  of  friction  may  vary 
from  .25  to  i.oo;  a  fair  general  value  being  about  .40.  In  this 
connection,  Mr.  Lewis  says:  "This  extreme  variation  in  the 
coefficient  of  friction  does  not  give  rise,  as  might  at  first  be  sup- 
posed, to  such  a  great  difference  in  the  transmission  of  power.  It 
will  be  seen  by  reference  to  formula  (i)  that  the  power  trans- 
mitted for  any  given  working  strength  and  speed  is  limited  only 
by  the  value  of  C,  which  depends  upon  the  arc  of  contact  and  the 
coefficient  of  friction.  For  the  usual  arc  of  contact  180°,  the 
power  transmitted  when  /*  =  .25  is  about  24  per  cent,  less  than 
when  /j.=  .40  and  when  /*=  i  oo,  the  power  transmitted  is  about 
33  per  cent,  more,  from  which  it  appears  that  in  extreme  cases 
the  power  transmitted  may  be  ^  less  or  ^  more  than  will  be 
found  from  the  use  of  Mr.  Nagle's  coefficient  of  .40.° 

The  paper  by  Mr.  Taylor,  referred  to  above,  gives  an  account 
of  "  A  Nine  Years'  Experiment  on  Belting,"  that  is  a  record  of 
careful  observations  and  measurements  for  nine  years  on  belts  in 
actual  use  at  the  works  of  the  Midvale  Steel  Co  Many  valuable 
facts  and  practical  suggestions  are  contained  in  this  paper,  but  a 
satisfactory  abstract  of  it  is  not  possible  in  this  place.  Mr.  Tay- 
lor advocates  thick  narrow  bells,  rather  than  thin  wide  belts.  He 
sums  up  his  investigation  in  36  "Conclusions";  the  first  of 
which  is  that : 

"  A  double  belt  having  an  arc  of  contact  of  180°,  will  give  an 
effective  pull  on  the  face  of  the  pulley  per  inch  of  width  of  belt 
of"  35  Ibs.  for  oak  tanned  and  fulled  leather  belts,  or  30  Ibs.  for 
other  types  of  leather  belts  and  6  to  7  ply  rubber  belts. 

"The  number  of  square  feet  of  double  belt  passing  around  a 
pulley  per  minute  required  to  transmit  one  horse-power  is"  80 
sq.  feet  for  the  oak  tanned  belt,  or  90  sq.  feet  for  other  leather 
belts  and  6  to  7  ply  rubber  belts. 

"  The  number  of  lineal  feet  of  double  belting  i  inch  wide  pass- 
ing around  a  pulley  per  minute  required  to  transmit  one  horse- 


-174  — 

power  is  "   950  feet  for  the  oak  tanned  belt ;  or  1,100  feet  for  the 
other  types  as  above. 

These  conclusions  are  based  upon  the  cost  of  maintaining  the 
belts  in  good  condition,  including  loss  of  time  from  repairs,  as 
well  as  other  considerations.  Smaller  values  than  these  rules 
dictate  are  generally  used,  because  of  the  first  cost  their  applica- 
tion would  involve. 

123.  Joints   in  Belting.     [Unwin,  §  252].     Belts  are  not  in- 
frequently made  without  any   laced  joint.     In   belted  dynamos, 
the  belt  is  tightened  by  moving  the  machine  on  its  bed  plate  ; 
suitable  adjusting    screws  being  provided  for  the   purpose.     In 
other  cases,  a  tightening  pulley  is  provided  ;  and,  again,  the  belt 
may  be  cemented  by  a  scarf  joint,    and  a  new  joint  be  made  if 
it  becomes  necessary  to  tighten   the  belt.     Small  belts  usually 
have  a  laced  joint.     If  a  tightener  is  used  it  should  run  against 
the  slack  side  of  the  belt,  and  it  is  usually  best  to  place  it  near 
the  smaller  pulley  to  increase  its  arc  of  contact  rather  than   that 
of  the  larger  pulley. 

124.  Cotton  Belting.     [Unwin,  §  253].     Cotton  belting  with 
a  thin  leather  contact  facing  has  been  used  to  a  considerable   ex- 
tent in  this  country. 

125.  Leather  Link  Belting.     [Unwin,  §254]. 

126.  Proportions  of  Pulleys.     [Unwin,  §§  258  to  263,  inclu- 
sive]. 


-  175  - 

XI.. 
ROPE  TRANSMISSION. 


127.  Transmission    Ropes    and    Sheaves.     [Unwin,    Page 
404  to  §  265,  and  also  §  269], 

128.  Strength  of  Ropes.   [Unwin,  §  265].     The  working  stress 
of  1 200  Ibs    per  square  inch  for  hemp  ropes,  as  assigned  by  Pro- 
fessor Unwin,  seems  to  be  much  in  excess  of  that  found  desirable 
by  experience.     Durability    of  ropes    demands  a  much  smaller 
working  stress,  relatively  to  the  ultimate  strength,  than  would  be 
provided  by  a  factor  of  safety  of  8  in  a  new  rope. 

Mr.  C.  W.  Hunt,  Past  President  of  the  American  Society  of 
Mechanical  Engineers,  presented  the  conclusions  reached  from 
his  extensive  experience  with  rope  transmission  in  a  paper  before 
the  Society,  (Transactions  Vol.  XII,  page  230).  An  abstract  of 
this  paper  is  given  in  art.  131  of  these  Notes. 

The  student  is  referred  to  the  work  in  Umvin's  Machine 
Design  for  the  general  theory  of  rope  transmission  ;  but  the 
formulas  and  data  given  in  Mr.  Hunt's  paper  are  recommended 
for  use  in  applications. 

129.  Driving  Force    and  Power    Transmitted  by  Ropes. 
[Unwin,  §§  266,  267]. 

130.  Friction    of    Ropes    in     Grooved     Sheaves.  [Unwin, 
§  268]. 

131.  Manilla  Rope  Transmission.     The  following  is,  in  the 
main,   an  abstract  of  Mr.    Hunt's  paper  before  the  A.  S.  M.  E., 
to  which  reference  was  made  in   art.    128;  the  quotations  being 
his  own  words  :   "  The  most  prominent   questions  which  the  en- 
gineer wishes  to  have  answered  who  proposes  to  make  an  appli- 
cation of  rope  driving  are  those  relating  to — 

Horse-power, 
Wear  of  rope, 
First  cost  of  rope, 
Catenary. 


-I76- 

These  questions  cannot  be  answered  with  precision  in  a  general 
article,  but  it  is  the  purpose  of  this  paper  to  give  the  gen- 
eral limitations  of  this  method  of  transmitting  energy." 

"  In  many  of  the  earlier  applications  so  great  a  strain  was  put 
upon  the  rope  that  the  wear  was  rapid,  and  success  only  came 
when  the  work  required  of  the  rope  was  greatly  reduced.  The 
strain  upon  the  rope  has  been  decreased  until  it  is  approximately 
known  what  it  should  be  to  secure  reasonable  durability." 
Experience  indicates  that  a  tension  in  the  driving  side  of  the  rope 
equivalent  to  200  Ibs.  on  a  manila  rope  i  inch  in  diameter  is  safe 
and  economical. 

Tests  of  ropes  from  different  makers  showed  an  average  break- 
ing strength  equivalent  to  7,140  pounds  on  a  rope  one  inch  in 
diameter. 

The  following  notation,  used  throughout  this  article,  has  been 
changed  to  correspond  to  that  used  by  Professor  Unwin. 

y  =  Circumference  of  rope  in  inches. 

8  =  S  ig  of  rope  in  inches. 

C=  Centrifugal  force  in  pounds. 

H.  P.  =  Horse-power  transmitted. 

/,=  Distance  between  pulleys  in  feet. 

w=  Weight  of  rope  in  Ibs.  per  foot  of  length. 

P=  Effective  force  (or  tension)  in  the  rope  in  Ibs. 

T^  =  Tension  on  driving  side  of  rope  in  Ibs.  ' 

7;=        "         "   slack         "    "     "     "     " 

v  =  Velocity  of  rope  in  feet  per  second. 

f=  Working  stress  in  one  rope. 

F^  Breaking  strength  of  one  rope. 

According  to  Mr.  Hunt  : 

F=720y2  (i) 

f=20y'i  (2) 

^=.0327*  (3) 

Since  F=  36/1  the  apparent  factor  of  safety  is  36.  The  work- 
ing stress  is  about  one  twenty-fifth  the  effective  strength  of  a  new 
rope,  allowing  for  the  splice.  "The  actual  strains  are  ordinarily 
much  greater,  owing  to  the  vibrations  in  running,  as  well  as 
from  imperfectly  adjusted  tension  mechanism." 


19 


•yaMOd  3SHOH 


The  maximum  working  stress  per  rope  is  taken  as  equivalent 
to  200  Ibs.  on  the  driving  side  of  a  i  inch  rope  (/=  207*);  and 
the  range  of  velocity  is  from  10  to  140  feet  per  second. 

The  centrifugal  action  produces  a  tension  of 

r        wv  l       .01,2  y*v2       y1  v  2     , 

C=    -_=—•?_'  ------  —  '.  —        (nearly)  (4) 

g  g  looo 

If  v  -~  141,  C=  2oy2—  /,  hence  the  centrifugal  tension  equals 
the  allowed  working  stress,  and  the  effective  pull  becomes 
zero  ;  that  is  no  power  can  be  transmitted  at  this  speed  without 
exceeding  the  assigned  maximum  tension  in  the  rope. 

Mr.  Hunt  states  that  there  have  been  no  experiments  which 
accutately  determine  the  coefficient  of  friction  of  lubricated  ropes 
on  pulleys  ;  but  that  "  when  a  rope  runs  in  a  groove  whose  sides 
are  inclined  toward  each  other  at  an  angle  of  45°  there  is  suffi- 
cient adhesion  when  " 

T,+  Tt  =  *  (5) 

However,  he  assumes  a  somewhat  different  ratio  of  T2  to  Tt  hi 
the  following  work,  viz:  "  That  the  tension  on  the  slack  side 
necessary  for  giving  adhesion  is  equal  to  one-half  the  force  doing 
useful  work  on  the  driving  side  of  the  rope."  Both  sides  of  the 
rope  have  a  component  of  tension  due  to  the  centrifugal  action  = 
C.  If  the  tension  for  adhesion  be  called  /„,  7~,  =  /0+  C,  and  Tt 
=  P+  t0  +  C;  but  if  „'„  be  taken  as  %  P,  Tl  =  %-P  +  C,  and 
T2  =  P+  Yz  P+  C=\P  +  C.  From  these  relations, 

p=2(7\_C}  ,6 

3 
/„  =  #/>=     r'~C  (7) 

O 

T,  =  t0+  C=y*P+  C=  ZLll^  +  C  (8) 

•  3 

From  the  above  relations, 


that  is,  the  ratio  of  the  total  tensions  on  the  two  sides  of  the  rope 
varies  with  the  effective  pull  and  with  the  speed  of  rope.  With 
the  assumption  that  the  tension  for  adhesion  is  one-  half  the 
effective  pull  (/„  =  %P\  Tt-^T,  equals  2  when  P=  2  C.  This 


-I78- 

correspouds  to  a  velocity  of  70.7  feet  per  second.  With  a  velocity 
of  80  feet  per  second,  which  is  about  the  velocity  for  maximum 
power  transmitted,  7!2  -r-  7]  =1.83. 

It  follows  from  eq.  (8)  tiiat  (T,  +  7",)  =  (4  Tt  —  2  C)  ;  or  tliat 
the  sum  of  the  tensions  is  not  constant  for  all  speeds.  This  last 
conclusion  is  arrived  at  without  reference  to  such  peculiar  rela- 
tions between  stress  and  strain  as  those  discussed  in  art.  122.  It 
is  probable  that  ropes,  like  belts,  undergo  decreasing  increments 
of  strain  with  increasing  increments  of  stress,  and  that  the  expres- 
sion for  (7^2  +  7])  should  be  modified  accordingly. 

"As  C  varies  as  the  square  of  the  velocity,  there  is,  with  -in 
increasing  speed  of  rope,  a  decreasing  useful  force,  and  an  in- 
creasing total  tension,  7]  ,  on  the  slack  side,"  for  a  fixed  value 
ofT,. 

With  UK-  preceding  assumptions,  the  horse-power  transmitted 
will  be  : 


550  3  X  550  825 

Transmission    ropes    are    usually    from    one    to    one    and   three- 
quarters  inches  in  diameter." 

Fig.  74  is  a  diagram  showing  the  horse  power  transmitted  by 
four  common  sizes  of  rope,  based  upon  a  total  tension  on  the 
driving  side  '  equivalent  to  200  Ibs.  on  a  one  inch  rope;  or 
7^=2oy?.  The  electrotype  for  this  diagram  was  kindly  fur- 
nished by  The  C.  W.  Hunt  Company. 

If  the  value  of  C  as  per  eq.  (4)  be  substituted  in  eq.  (9)  and  the 
resulting  expression  be  solved  for  a  maximum,  it  will  be  found 
that  the  greatest  horse-pow^r  is  transmitted  at  a  velocity  of  about 
82  feet  per  second.  An  inspection  of  Fig.  74  shows  its  agree- 
ment with  this  result.  In  order  that  the  value  of  T.t  shall  be 
maintained  equal  to  20  y'2,  the  effective  pull  must  be  reduced  as 
the  centrifugal  force  is  increased.  The  energy  transmitted  equals 
Pv,  and  if  v  exceeds  80  feet  per  second  P  must  be  reduced  at  a 
greater  rate  than  v  increases,  on  account  of  the  rate  at  which  the 
centrifugal  force  is  augmented. 

If  Tt  is  increased  with  the  speed,  greater  power  may,  of  course, 
be  transmitted  at  a  higher  velocity,  and  the  first  cost  is  reduced  ; 


-  179  — 

but  this  is  done  at  the  expense  of  life  of  the  rope.  With  a  fixed 
value  of  7"2  =  20  y2,  the  first  cost  is  a  minimum  at  about  v  =  So, 
and  this  first  cost  is  greater  by  about  10  per  cent,  if  v  is  increased 
to  100  or  decreased  to  62  feet  per  second.  The  first  cost  is  in- 
creased by  50  per  cent,  when  the  velocity  is  reduced  to  40  feet 
per  second. 

"  The  wear  of  rope  is  both  internal  and  external ;  the  internal 
is  caused  by  the  movement  of  the  fibres  on  each  other,  under 
pressure  in  bending  over  the  sheaves,  and  the  external  is  caused 
by  the  slipping  and  wedging  in  the  grooves  of  the  pulley.  Both 
of  these  causes  of  wear  are,  within  the  limits  of  ordinary  practice, 
assumed  to  be  directly  proportional  to  the  speed." 

Equation  (9)  shows  -that  the  power  transmitted  does  not  vary 
at  this  same  rate  "The  higher  the  speed,  up  to  about  80  feet  per 
second,  the  more  power  will  be  transmitted,  but  it  is  accompanied 
by  more  than  equivalent  wear." 

The  smallest  sheave  over  which  a  rope  runs  in  a  transmission 
should  have  a  diameter  not  less  than  about  40  times  the  diameter 
of  the  rope. 

"  There  are  two  methods  of  putting  ropes  on  the  pulleys  ;  one 
in  which  the  ropes  are  single  and  spliced  on,  being  made  very 
taut  at  first,  and  less  so  as  the  rope  lengthens,  stretching  until  it 
slips,  when  it  is  respliced.  The  other  method  is  to  wind  a  single 
rope  over  the  pulley  as  many  times  as  is  needed  to  obtain  the 
necessary  horse-power  and  put  a  tension  pulley  to  give  the  neces- 
sary adhesion  and  also  to  take  up  the  wear  [stretch]."  This  pulley 
is  also  necessary  to  convey  the  rope  to  the  main  pulleys  in  the 
proper  planes. 

The  catenary  or  sag  of  the  tight  side  of  the  rope  is  constant  at 
all  speeds  if  the  tension  on  this  side  is  constant.  "  The  deflection 
of  the  rope  between  the  pulleys  on  the  slack  side  varies  with  each 
change  of  the  load  or  change  of  speed,  as  the  tension  equation 
(8)  indicates." 

The  following  formula  may  be  used  for  computing  the  approxi- 
mate sag  in  inches,  taking  T  as  the  tension  on  the  side  of  the 
rope  under  consideration  ;  that  is,  T—  7\  for  the  sag  of  the  tight 
side,  or  T=  7J  (as  given  by  eq.  8)  for  the  sag  of  the  slack  side, 


—  i8o 


—  —        -  ( 10") 

o  «""  V  * v-'/ 

2  W          -^    4  Z£>" 

Great  care  is  necessary  in  the  construction  of  a  rope  sheave  to 
have  all  of  the  grooves  of  same  depth  and  form.  Neglect  of 
this  point  will  result  in  excessive  tension  on  some  of  the  ropes, 
while  others  are  subjected  to  much  lower  tension'. 

The  pitch,  or  effective,  diameter  of  a  rope  sheave  is  the  diameter 
measured  to  the  center-line  of  the  rope,  or  to  the  neutral  axis  of 
the  rope  which  wraps  around  the  wheel.  The  effective  diameter 
is  equal  to  the  length  of  rope  required  to  reach  once  around  the 
wheel  divided  by  TT  ;  this  length  of  rope  being  taken  when  it  is 
under  the  working  stress.  If  two  parallel  ropes  connect  two 
sheaves,  imagine  the  grooves  of  the  driven  wheel  to  be  exactly 
alike,  but  that  one  of  the  grooves  of  the  driver  has  a  larger 
effective  diameter  than  the  other.  The  linear  velocity  of  the  rope 
running  to  the  groove  of  larger  pitch  diameter  will  be  greater 
than  that  of  the  parallel  rope  ;  hence  it  will  tend  to  carry  all  of 
the  load,  with  a  corresponding  increase  of  its  tension  and  diminu- 
tion of  the  tension  on  the  other  rope. 

132.  Wire  Rope  Transmission.  [Unvvin,  §§270  to  277]. 
The  recent  development  of  systems  of  electrical  transmission  of 
power  has  greatly  curtailed  the  field  of  wire  rope  transmission  ; 
though  there  are  conditions  under  which  wire  rope  may  still  be 
advantageously  employed. 

The  table  on  the  following  page,  which  is  taken  from  a  circular 
of  the  John  A.  Roebling's  Sons  Co.,  shows  the  power  that  may 
be  transmitted  by  ropes  of  various  sizes  with  sheaves  of  different 
diameters  and  rotative  speeds.  These  values  are  for  a  rope  made 
with  six  strands  around  a  hemp  core,  each  strand  consisting  of 
seven  wires.  This  table  does  not  make  allowances  for  the 
change  of  stress  due  to  change  of  centrifugal  force  at  various 
speeds  ;  but  it  does  consider  the  influence  of  the  sheave  diameter 
on  the  bending  stress.  For  example  :  a  $/%'  rope  on  an  eight  foot 
sheave  running  100  i .  p.  m.  transmits  only  32  H.  P.;  while  the 
same  rope  transmits  64  H.  P.  when  running  on  a  ten  foot  sheave 
at  80  revs,  per  minute,  or  at  the  same  linear  velocity. 


—  181  — 
TABLE  OF  TRANSMISSION  OF  POWER  BY   WIRE  ROPES. 


Diameter  of 
wheel  in  feet. 

Number  of 
revolutions 

Trade  Number 
ofrope. 

Diameter  of 
rope. 

Horse-Power 

Diameter  of 
wheel  in  feet. 

Number  of 
revolutions. 

Trade  Number 
of  rope. 

Diameter  of 
rope. 

K 

3 

80 

23    j    X 

3 

7 

140             20 

A 

35  . 

3 

IOO 

23      y* 

3/2 

8 

80             19 

ft         26 

3 

1  20 

23 

H 

4 

8 

IOO                   19         ft                   32 

3 

140 

23 

H 

4% 

8 

120             19       #             39 

4 

80 

23 

H 

4            8 

140 

19       ft             45 

4 

IOO 

23 

H 

5 

9 

80 

f  20 

}T9s  ft 

{   48 

4 

120 

23        X 

6            9 

(  20 

}T°*  ft 

{    6^ 

4 
5 

140 
80 

23 

22 

H 
A 

7 
9 

9 

Q 

1  20 
140 

(  20 

U9 

{i 

}T9*  ft 
}r9*  ft 

Si 

5 

IOO 

22                   T~5 

ii 

IO 

80 

}ft  H 

{   68 

5 

120 

22          rV 

13 

IO 

IOO 

A 

}ft  H 

{   8^ 

5 

140 

22 

A 

'5 

10 

120 

{3 

}ftH 

J    96 

\  102 

6 

80 

21 

^ 

H 

10 

140 

/i9 
\i8 

}ftU 

(112 
\II9 

6 
6 

IOO 

1  20 

21 
21 

* 

20 

12 
12 

80 
IOO 

{£ 
{!? 

}»* 

|   93 

(  116 
\  124 

6 

140 

21 

a 

23 

12 

I2O 

d 

W 

(140 
\I49 

7 

80 

20 

A 

20 

12 

120                  l6 

ft 

173 

7             loo 

2O 

25 

14 

80          {    J 

7            120 

20 

,: 

30 

'4    , 

IOO             {     « 

I1  'MIS 

*  Taken  from  a  publication  of  the  John  A.  Roebling's  Sons  Company,  of 
Trenton,  N.  J. 

The  above  table  gives  the  power  produced  by  Patent  Rubber-lined  Wheels 
and  Wire  Belt  Ropes,  at  various  speeds. 

Horse-powers  given  in  this  table  are  calculated  with  a  liberal  margin  for 
any  temporary  increase  of  work. 


-  182  — 

For  hoisting,  and  for  transmission  if  the  sheave  diameters  must 
he  much  smaller  than  those  given  in  the  preceding  table,  a  more 
flexible  rope  is  used.  This  consists  of  six  strands  around  a  hemp 
core,  but  each  strand  is  made  up  of  nineteen  wires,  which  are,  of 
course,  of  smaller  diameter  than  those  used  for  corresponding 
sizes  of  seven-wire  rope.  The  lining  of  the  bottoms  of  the 
grooves  in  the  sheaves  should  be  maintained  in  good  repair.  If 
it  becomes  irregular,  through  wear,  the  rope  may  be  bent  at  a 
sharp  angle  in  passing  over  the  high  spots  of  the  lining,  with  a 
resultant  increase  in  the  stress  of  the  wires.  This  last  action  is 
not  equivalent  to  running  over  a  correspondingly  smaller  sheave, 
however,  for  every  portion  of  each  wire  is  bent  around  each 
sheave  once  during  every  circuit  of  the  rope  ;  while  it  is  not 
likely  that  the  same  portion  of  the  rope  will  frequently  come  in 
contact  with  any  irregularity  in  the  lining. 

133.  The    Catenary  and   Sag   of  Rope.     [Unwin,    §   285.] 
The  approximate  equations  of  §   285   (Unwin)   are  sufficiently 
exact  for  most  problems  of  practice,  and  the  discussion  of  §§  278 
to  285  will  be  omitted. 

134.  Efficiency    of   Wire    Rope    Transmission.      [Unwin, 
§  286.] 

135.  Pulleys  for  Wire  Rope  Transmission.    [Unwin,  §  287.] 

136.  Velocity,  Wear,  Stretch,  etc.     [Unwin,  §§  288  to  290, 
inclusive.] 


—  i83  — 
XII. 

CHAINS  AND  CHAIN  WHEELS. 


137.  Chains  used  for  Cranes,  etc.     [Unwin,  §§  291  to  299, 
inclusive.] 

138.  Chains  and   Sprocket  Wheels  for  Transmission    of 
Power.     [Unwin,  §§  300  to  304.] 

See  also  Kinematics  of  Machinery,  art.  121,  pages  214-215. 


This  book  is  DUE  on  the  last  date  stamped  belc 


Form  L-9-15m-7,'32 


TJ 

230  Barr  - 
U62eb  Notes  on 

the  design 
of  machine  elements 


A    001245902    o 


'-•T.TFORNJJk 


'8IUBI 


